1 Bilinear Forms

A Review of Linear Algebra.

Definition 1.1. • A square matrix P of real componentsis said to be an orthogonal matrix if tPP = P tP = Iholds, where tP denotes the transposition of P and I isthe identity matrix.

• A square matrix A is said to be (real) symmetric matrixif tA = A holds.

Fact 1.2. • The eigenvalues of a real symmetric matrix arereal numbers, and the dimension of the corresponding eigenspacecoincides with the multiplicity of the eigenvalue.

• Real symmetric matrices can be diagonalized by orthogonalmatrices. In other words, for each real symmetric matrixA, there exists an orthogonal matrix P satisfying

P−1AP = tPAP = diag(λ1, . . . , λn),

where diag(. . . ) denotes the diagonal matrix with diago-nal components “. . . ”. In particular, {λ1, . . . , λn} are theeigenvalues of A counted with their multiplicity.

In this section, V denotes an n-dimensional vector space overR (n <∞).

09. April, 2019. Revised: 16. April, 2019

MTH.B405; Sect. 1 (20190721) 2

Bilinear forms and quadratic forms.

Definition 1.3. A symmetric bilinear form on the vector spaceV is a map q : V × V → R satisfying the following:

• For each fixed x ∈ V , both

q(x, ·) : V ∋ y 7→ q(x,y) ∈ R and

q(·,x) : V ∋ y 7→ q(y,x) ∈ R

are linear maps.

• For any x and y ∈ V , q(x,y) = q(y,x) holds.

The quadratic form associated to the symmetric bilinear form qis a map q : V ∋ x 7→ q(x,x) ∈ R.

Lemma 1.4. A quadratic form determines the symmetric bi-linear form. In other words, two symmetric bilinear forms withcommon quadratic form coincide with each other.

Proof. Let q be a symmetric bilinear form and q the quadraticform associated to it. Since

q(x+ y) = q(x+ y,x+ y)

= q(x,x) + q(x,y) + q(y,x) + q(y,y)

= q(x) + 2q(x,y) + q(y)

holds for each x, y ∈ V , we have

q(x,y) =1

2

(q(x+ y)− q(x)− q(y)

).

Hence the symmetric bilinear form q is determined by q.

3 (20190721) MTH.B405; Sect. 1

By virtue of Lemma 1.4, a symmetric bilinear form itself isoften called a quadratic form.

Example 1.5. For an n× n symmetric matrix A = (aij) withreal components and column vectors x, y ∈ Rn, we set

(1.1) qA : Rn × Rn ∋ (x, y) 7−→ txAy ∈ R,

where tx the column vector obtained by transposing x. ThenqA is a symmetric bilinear form on Rn. In particular, qI is thecanonical inner product of Rn, where I is the identity matrix.

Conversely, for each symmetric bilinear form q in Rn, thereexists a symmetric matrix A such that q = qA. In fact, settingaij := q(ei, ej), A = (aij) satisfies q = qA, where [ej ] is thecanonical basis of Rn.

Matrix representation of quadratic forms. Take a basis[v1, . . . ,vn] of the vector space V . Then

(1.2) V ∋ x 7−→ x =

x1...xn

∈ Rn,

x = x1v1 + · · ·+ xnvn = [v1, . . . ,vn]

x1...xn

gives an isomorphism of vector spaces. We call x ∈ Rn thecomponent of x with respect to the basis [vj ].

MTH.B405; Sect. 1 (20190721) 4

Lemma 1.6. For a symmetric bilinear form q on V , there existsthe unique n× n symmetric matrix A satisfying

q(x,y)

(= qA(x, y)

)= txAy,

where x and y ∈ Rn are the components of x, y with respect to[v1, . . . ,vn], respectively

Proof. Setting A = (aij) by aij := q(vi,vj), the conclusionfollows. In fact,

q(x,y) = q

n∑

i=1

xivi,n∑

j=1

yjvj

=

n∑

i,j=1

xiyjq(vi,vj)

=n∑

i,j=1

xiyjaij =txAy = qA(x, y),

where x =t(x1, . . . , xn) and y =

t(y1, . . . , yn).

We call the matrix A in Lemma 1.6 the representative matrixof q with respect to the basis [vj ].

Lemma 1.7. Take two bases [v1, . . . ,vn], [w1, . . . ,wn] of Vand let U = (uij) ∈ GL(n,R) be the basis change matrix:

[w1, . . . ,wn] = [v1, . . . ,vn]U

i.e., wj =n∑

i=1

uijvi (j = 1, . . . , n),

5 (20190721) MTH.B405; Sect. 1

where GL(n,R) denotes the general linear group, that is, the setof n × n regular matrix of real components. Let A (resp. A) bethe representative matrix of a symmetric bilinear form q withrespect to the basis [vj ] (resp. [wj ]). Then it holds that

A = tUAU.

Proof. Writing x and y as

x = [v1, . . . ,vn]x = [w1, . . . ,wn]x,

y = [v1, . . . ,vn]y = [w1, . . . ,wn]y,

we havex = U x, y = U y.

Hence q(x,y) = txAy = txtUAU y.

Non-degenerate quadratic forms.

Definition 1.8. A symmetric bilinear form (a quadratic form)q on V is said to be

• positive definite (resp. positive semi definite) if q(x,x) > 0(resp. ≧ 0) holds for all x ∈ V \ {0},

• negative definite if −q is positive definite,

• non-degenerate when “q(x,y) = 0 for all y ∈ V ” implies“x = 0”.

Example 1.9. An inner product (in the undergraduate LinearAlgebra course) is nothing but a positive definite quadratic form.

MTH.B405; Sect. 1 (20190721) 6

Remark 1.10. A positive (resp. negative) quadratic form is non-degenerate. In fact, if q(x,y) = 0 holds for all y, q(x,x) = 0holds. On the other hand, q(x,x) > 0 (resp. < 0) when x = 0.Hence x = 0.

Signature of non-degenerate quadratic forms.

Proposition 1.11. A quadratic form q on V is positive definite(resp. positive semi-definite, negative definite, non-degenerate)if and only if all eigenvalues of the representative matrix of q arepositive (resp. non-negative, negative, non-zero). This conditiondoes not depend on choice of bases.

Proof. Let [v1, . . . ,vn] be a basis of V and A the representativematrix of q with respect to it. Since A is a symmetric matrix,there exists an orthogonal matrix P such that

tPAP = Λ, Λ :=

λ1 . . . 0...

. . ....

0 . . . λn

,

tPP = I = the identity matrix,

where λ1,. . . , λn are eigenvalues of A, which are real numbers.Then, by setting [w1, . . . ,wn] := [v1, . . . ,vn]P , the representa-tive matrix of q with respect to [wj ] is the diagonal matrix Λ.Denoting the components of vectors x,y with respect to [wj ] by

x =t(x1, . . . , xn) and y =

t(y1, . . . , yn), respectively,

q(x,y) =

n∑

j=1

λjxjyj .

7 (20190721) MTH.B405; Sect. 1

The conclusion is obtained by this equality. In fact, if q is pos-itive (resp. negative) definite, q(wj ,wj) = λj is positive (resp.negative) for each j = 1, . . . , n. Hence all eigenvalues of A arepositive (resp. negative). Conversely, if all eigenvalues are posi-tive (resp. negative),

q(x,x) =n∑

j=1

λj(xj)2

is positive (resp. negative). The conclusion for positive semi-definite case is obtained in the same way.

On the other hand, let q be a non-degenerate quadratic formand assume λj = 0 for some j = 1, . . . , n. Then

q(wj ,wj) = λj = 0,

contradiction to non-degeneracy. Conversely, assume λj = 0(j = 1, . . . , n), and q(x,y) = 0 for all y ∈ V . Then

0 = q(x,wj) = λjxj

holds, which implies xj = 0, for j = 1, . . . , n. Thus, x = 0.bHence q is non-degenerate.

Let W be a linear subspace of the vector space V . Then asymmetric bilinear form q : V ×V → R on V induces a symmet-ric bilinear form q|W : W ×W → R on W .

Definition 1.12. For a non-degenerate quadratic form q on V ,we define

n+ := max{dimW ; q|W is positive definite},n− := max{dimW ; q|W is negative definite}.

MTH.B405; Sect. 1 (20190721) 8

The pair (n+, n−) is called the signature of q.

Example 1.13. A positive (resp. negative) definite quadraticform q on V has signature (n, 0) (resp. (0, n)), where n = dimV .

Theorem 1.14. Let (n+, n−) be the signature of a non-degeneratequadratic form q on V . Then n+ (resp. n−) is the number ofpositive (resp. negative) eigenvalues of the representative matrixof q. In particular, n+ + n− = n = dimV holds.

Proof. As seen in the proof of Proposition 1.11 we may assumethat the matrix representative with respect to the basis [wj ] is adiagonal matrix Λ, without loss of generality. Since all diagonalcomponents of Λ are non-zero, we may assume that λ1, . . . , λtare negative, and λt+1, . . . , λn are positive. Then q is negativedefinite on the subspace generated by {w1, . . . ,wt}, and hencen− ≧ t. On the other hand, q is positive definite on the subspacegenerated by {wt+1, . . . ,wn}, and then n+ ≧ n− t:

(1.3) n− ≧ t, n+ ≧ n− t.

Here, by definition, there exists a subspace W+ (resp. W−)of V such that q|W+

(resp. q|W−) is positive (resp. negative)definite and dimW+ = n+ (resp. dimW− = n−). Take a vectorx ∈ W+ ∩W−. Then q(x,x) ≦ 0 and q(x,x) ≧ 0 hold, that is,q(x,x) = 0. Noticing q|W+ is positive definite, x = 0. HenceW+ ∩W− = {0}, and then we have

n+ + n− = dimW+ + dimW− ≦ dimV = n.

Therefore (1.3) yields

n− ≧ t, n− n− ≧ n− t n− n+ ≧ t, n+ ≧ n− t,

9 (20190721) MTH.B405; Sect. 1

that is, n− = t, n+ = n− t.

Remark 1.15. By Theorem 1.14, the number of positive (resp.negative) eigenvalues of the matrix representative does not de-pend on choice of bases. This fact is equivalent to “the numberof positive (negative) eigenvalues of a symmetric matrix A isinvariant to the transformation A 7→ tUAU (U is a regular ma-trix)”.

Definition 1.16. An inner product on a finite dimensional vec-tor space V is a non-degenerate quadratic (symmetric bilinear)form. A vector space with fixed inner product is called an innerproduct space or a metric space.

Pseudo Euclidean vector spaces. Let s ≧ 0, t ≧ 0 beintegers satisfying n := s+ t ≧ 2. Then a quadratic form

(1.4) ⟨v,w⟩s,t := −

s∑

j=1

vjwj

+

s+t∑

k=s+1

vkwk

v =

v1...vn

,w =

w1

...wn

gives an inner product on Rn with signature (t, s). We denote byRn

s such an inner product space, and call the pseudo Euclideanvector space of signature (s, t). The inner product (1.4) can be

MTH.B405; Sect. 1 (20190721) 10

expressed as

(1.5) ⟨v,w⟩s,t = tvJs,tw Js,t :=

(−Is OO It

).

In particular, the case of signature (n, 0), Rn := Rn0 is called

the Euclidean vector space, and when the signature is (n−1, 1),the space Rn

1 is called the Minkowski vector space.

Orthonormal basis In this paragraph, we fix an inner prod-uct ⟨ , ⟩ on V .

Definition 1.17. A vector v ∈ V is said to be orthogonal tow ∈ V if ⟨v,w⟩ = 0 holds.

Definition 1.18. An n-tuple {e1, . . . , en} of V is called anorthonormal basis of V if

| ⟨ei, ej⟩ | = δij (1 ≦ i, j ≦ n)

holds.

Lemma 1.19. An orthonormal basis is a basis of V .

Proof. It is sufficient to show linear independency.

Theorem 1.20. There exists an orthonormal basis for an arbi-trary inner product space. In particular, if the signature of theinner product is (t, s), one can take a basis [ej ] satisfying

⟨ei, ej⟩ = 0 (i = j),

⟨ei, ei⟩ ={−1 (i = 1, . . . , s)

1 (i = s+ 1, . . . , s+ t).

11 (20190721) MTH.B405; Sect. 1

Proof. As seen in the proof of Proposition 1.11, there exists abasis [wj ] such that the matrix representative of ⟨ , ⟩ is an diag-onal matrix Λ = diag(λ1, . . . , λn). Since the number of positive(resp. negative) eigenvalues is t (resp. s), we may assume

λj

{< 0 (j = 1, . . . , s)

> 0 (j = s+ 1, . . . , n)

without loss of generality. We set

U := diag(1/√|λ1|, . . . , 1/

√|λn|

).

Then it holds that

(1.6) tUΛU =

(−Is OO It

),

where Im is the m×m identity matrix, and O denotes the zeromatrix of an appropriate size. Hence by Lemma 1.7, the matrixrepresentative of ⟨ , ⟩ with respect to the basis [e1, . . . , en] :=[w1, . . . ,wn]U is the matrix in (1.6). Hence [ej ] satisfies thedesired property.

MTH.B405; Sect. 1 (20190721) 12

Exercises

1-1 For an m × n matrix C, we set A := tCC, which is ann×n-symmetric matrix. Let qA be the quadratic form onRn as in Example 1.5 induced from A.

(1) Prove that qA is positive semi-definite.

(2) Find a condition of C for qA to be positive definite.

1-2 Let M2(R) be the set of 2× 2 real matrices, and

Sym(2,R) := {A ∈ M2(R) ; tA = A},Sym+(2,R) := {A ∈ Sym(2,R) ; qA is positive definite},

where qA is the quadratic form as defined in Example 1.5.Is the subset Sym+(2,R) a smooth submanifold of M2(R) =R4?

13 (20190721) MTH.B405; Sect. 2

2 Inner product spaces

Fix an inner product ⟨ , ⟩ of signature (n−, n+) on an n-dimensionalvector space V , where n = n+ + n−.

Orthogonal transformations. A map f : V → V is an or-thogonal transformation of the inner product space (V, ⟨ , ⟩) if

⟨f(x), f(y)⟩ = ⟨x,y⟩holds for all x, y ∈ V .

Lemma 2.1. An orthogonal transformation is a bijection.

Proof. Let f : V → V be an orthogonal transformation. Takex, y ∈ V satisfying f(x) = f(y). Then

⟨f(x)− f(y),w⟩ = 0, that is, ⟨f(x),w⟩ = ⟨f(y),w⟩holds for an arbitrary w ∈ V . In particular, setting w = f(z),we have ⟨f(x), f(z)⟩ = ⟨f(y), f(z)⟩ and then

⟨x, z⟩ = ⟨y, z⟩ , that is, ⟨x− y, z⟩ = 0

holds for all z ∈ V . Thus, by non-degeneracy of the innerproduct, we have x = y, that is, f is injective.

To show surjectivity, take an orthonormal basis {ej}nj=1 on(V, ⟨ , ⟩). By orthogonality, {f(ej)}nj=1 is also an orthonormalbasis. Then an arbitrary v ∈ V is expressed as

v =n∑

j=1

vjf(ej), vj := ⟨v, f(ej)⟩ (j = 1, . . . , n).

16. April, 2019. Revised: 23. April, 2019

MTH.B405; Sect. 2 (20190721) 14

Setting

x :=n∑

j=1

vjej , we have vj = ⟨x, ej⟩ = ⟨f(x), f(ej)⟩ .

Then f(x) = v holds. In fact,

f(x) =n∑

j=1

⟨f(x), f(ej)⟩ f(ej) =n∑

j=1

vjf(ej) = v.

Hence f is surjective.

Proposition 2.2. An orthogonal transformation is an linearisomorphism.

Proof. Take x, y ∈ V . For an arbitrary w ∈ V , there existsz ∈ V such that f(z) = w, because of Lemma 2.1. Then wehave

⟨f(x+ y),w⟩ = ⟨f(x+ y), f(z)⟩ = ⟨x+ y, z⟩= ⟨x, z⟩+ ⟨y, z⟩ = ⟨f(x), f(z)⟩+ ⟨f(y), f(z)⟩= ⟨f(x) + f(y),w⟩

for an arbitrary w ∈ V . Hence by non-degeneracy of ⟨ , ⟩,

f(x+ y) = f(x) + f(y)

holds. Similarly one can show that f(tx) = tf(x) for x ∈ Vand t ∈ R.

15 (20190721) MTH.B405; Sect. 2

Orthogonal transformations of Rns . Consider a pseudo Eu-

clidean space Rns ((n−, n+) = (s, n − s)), that is, the space Rn

with the inner product ⟨ , ⟩ := ⟨ , ⟩s,n−s as in (1.4). Recall thatthe inner product ⟨ , ⟩ is expressed as

(2.1) ⟨x,y⟩ = txJs,n−sy, Js,n−s :=

(−Is OO In−s

),

where Ip denotes the p×p-identity matrix, and O denotes a zeromatrix of an appropriate size. We write

(2.2) O(n, s) := {A ∈ GL(n,R) | tAJs,n−sA = Js,n−s},

which is identified with the set of orthogonal transformations ofRn

s . Obviously, O(n, s) is a group with respect to matrix multi-plications. We call this group the orthogonal group of signature(s, n− s). Decompose a matrix A ∈ O(n, s) as

(2.3) A =

(A11 A12

A21 A22

)

{A11 : s× s-matrix A12 : s× (n− s)-matrix

A21 : (n− s)× s-matrix; A22 : (n− s)× (n− s)-matrix.

Lemma 2.3. For each A ∈ O(n, s),

• detA = ±1,

• |detA11| ≧ 1, where A11 is the upper-left component as inthe decomposition (2.3).

MTH.B405; Sect. 2 (20190721) 16

Proof. Taking determinant of (2.2), we have the first assertion.If the matrix A as in (2.3) satisfies (2.2), the upper-left compo-nent satisfy

(2.4) −tA11A11 +tA21A21 = −Is.

Let λ ∈ R and x ∈ Rs \ {0} be an eigenvalue and the corre-sponding eigenvector of s× s-symmetric matrix tA11A11. Thenit holds that

λx = tA11A11x = (Is +tA21A21)x = x+ tA21A21x,

and hence

λ|x|2 = λtxx = |x|2 + txtA21A21x = |x|2 + |A21x|2 ≧ |x|2,where | · | denotes the Euclidean norm of Rs or Rn−s. Since|x| > 0, we have λ ≧ 1, that is, all eigenvalues of tA11A11 aregreater than or equal to 1. Hence

|detA11|2 = |det tA11A11| ≧ 1.

So we have the second assertion.

Notation 2.4. For a positive integer n and a non-negative in-teger s ≧ 0, we write

SO(n, s) := {A ∈ O(n, s) | detA = 1},which is a subgroup of O(n, s) (of index 2). Moreover, whenn ≧ 2 and 1 ≦ s ≦ n− 1, we set

SO+(n, s) := {A ∈ SO(n, s) | detA11 ≧ 1},where A11 is an s× s-matrix as in (2.3).

17 (20190721) MTH.B405; Sect. 2

Remark 2.5. Identifying the set M(n,R) of n× n real matrices

with the Euclidean space Rn2

, O(n, s) is a real analytic closedsubmanifold (cf. Section 3). Since the group operations (the ma-trix multiplication and the inversion) of O(n, s) are real analyticmaps from O(n, s) × O(n, s) to O(n, s) and O(n, s) to O(n, s),respectively, O(n, s) has a structure of a Lie group. Notice thatO(n, s) is not connected. In fact, the image of a continuous map

det : O(n, s) ∋ A 7−→ detA ∈ R

is {−1, 1} which is not connected.When s = 0, one can show SO(n) := SO(n, 0) is connected.

On the other hand, if 1 ≦ s ≦ n − 1, SO(n, s) consists of twoconnected components

SO+(n, s) :={A ∈ SO(n, s) ; detA11 ≧ 1},{A ∈ SO(n, s) ; detA11 ≦ −1}.

Orthogonal Complements. We let (V, ⟨ , ⟩) be an innerproduct space of dimension n with signature (n+, n−).

Definition 2.6. A linear subspaceW is said to be non-degenerateif the restriction ⟨ , ⟩ |W of the inner product on W is non-degenerate quadratic form on W . The signature of ⟨ , ⟩ |W iscalled the signature of W .

Example 2.7. A linear subspace of R21 spanned by

t(1, 1) is a

degenerate (i.e., not non-degenerate) subspace.

Definition 2.8. For a linear subspace W ⊂ V , we define

W⊥ := {v ∈ V | ⟨v,w⟩ = 0 for all w ∈W},

MTH.B405; Sect. 2 (20190721) 18

and call it the orthogonal complement of W .

Theorem 2.9. Let (n+, n−) be the signature of the inner prod-uct ⟨ , ⟩ of V and take a non-degenerate subspaceW of V . ThenV =W ⊕W⊥, and the signature of ⟨ , ⟩ |W⊥ is (n+−m+, n−−m−), where (m+,m−) is the signature of ⟨ , ⟩ |W .

Proof. Since W is non-degenerate, there exists an orthonormalbasis [e1, . . . , em] (m = m+ + m−) because of Theorem 1.20.Consider a liner map

Φ : V ∋ x 7−→ Φ(x) =

⟨x, e1⟩...

⟨x, em⟩

∈ Rm.

Since W = KerΦ, W⊥ is a linear subspace of V . Moreover, Φis surjective. In fact, for each x :=

t(x1, . . . , xm) ∈ Rm, we set

x := x1e1 + · · ·+ xmem. Then Φ(x) = x, and so

dimW⊥ = dimV − dim ImΦ = dimV −m = dimV − dimW.

We show thatW ∩W⊥ = {0}. In fact, take x ∈W ∩W⊥. Sincex ∈ W⊥, ⟨x,y⟩ = 0 for an arbitrary y ∈ W . Here, noticingx ∈ W , x = 0 because of non-degeneracy of W . Summing up,the sum of W and W⊥ is a direct sum, and

dim(W ⊕W⊥) = dimW + dimW⊥ = n = dimV

holds. Hence W ⊕W⊥ = V .

19 (20190721) MTH.B405; Sect. 2

Next, we show non-degeneracy of W⊥. For a fixed x ∈W⊥ \ {0}, define a linear map

Ψ: V ∋ y 7−→ Ψ(y) = ⟨x,y⟩ ∈ R.

Since ⟨ , ⟩ is non-degenerate on V , Ψ is not identically zero, inparticular, rankΨ = 1. Hence dimKerΨ = n− 1. On the otherhand, since W⊥ ⊂ KerΨ by definition,

dim(W⊥ ∩KerΨ) = dimW⊥ − 1.

This implies the existence of y ∈ W⊥ satisfying ⟨x,y⟩ = 0.Hence W⊥ is non-degenerate.

Finally, we compute the signature ofW⊥. Take orthonormalbases [e1, . . . , em] and [f1, . . . ,fn−m] of W and W⊥, respec-tively. Then [e1, . . . , em,f1, . . . ,fn−m] is an orthonormal basisof V , and the matrix representative of ⟨ , ⟩ is a diagonal matrixwhose diagonal components consist of 1 and −1. In particular,the number of 1’s of the diagonal components is the sum of thepositive signs ofW andW⊥, which equals to n+. Thus, we havethe conclusion.

Minkowski vector space. We consider a special case of sig-nature (n − 1, 1). Such an inner product space is called aMikowski vector space.

Definition 2.10 (Causality). A vector v in a Minkowski vectorspace V is said to be

• space-like if ⟨v,v⟩ > 0 or v = 0.

MTH.B405; Sect. 2 (20190721) 20

• time-like if ⟨v,v⟩ < 0, and

• light-like or null if ⟨v,v⟩ = 0 and v = 0.

These properties of v is called causality.

A space-like (resp. time-like) vector v is said to be a unitvector if | ⟨v,v⟩ | = 1.

For a non-zero vector v ∈ V , we set

v⊥ := (Rv)⊥ = {w ∈ V ; ⟨v,w⟩ = 0},

which is an (n− 1)-dimensional linear subspace of V because ofTheorem 2.9.

Proposition 2.11. For a vector v ∈ V \{0}, v⊥ is an (n− 1)-dimensional subspace. Moreover, it is

• a space-like subspace and V = Rv ⊕ v⊥ if v is time-like,

• a Minkowski subspace and V = Rv⊕v⊥ if v is space-like,and

• a degenerate subspace containing v if v is light-like.

Proof. The first two assertions are corollaries of Theorem 2.9.If v is light-like, v ∈ v⊥ by definition.

Take an orthonormal basis [e0, . . . , en−1] of V with ⟨e0, e0⟩ =−1. Then

Φ : V ∋ v 7−→

⟨v, e0⟩...

⟨v, en−1⟩

∈ Rn

1

21 (20190721) MTH.B405; Sect. 2

preserves the inner product. Thus, we can identify a Minkowskivector space V with Rn

1 . We write components of Rn1 as

x =t(x0, x1, . . . , xn).

Vector products of Minkowski vector spaces.

Definition 2.12. For each v =t(v0, v1, v2),w =

t(w0, w1, w2) ∈

R31, we set

v ×w :=t(

−∣∣∣∣v1 w1

v2 w2

∣∣∣∣ ,∣∣∣∣v2 w2

v0 w0

∣∣∣∣ ,∣∣∣∣v0 w0

v1 w1

∣∣∣∣)

and call it the vector product of v and w.

Lemma 2.13. For v, w ∈ R31,

• v ×w is perpendicular to both v and w,

• if both v and w are space-like, v ×w is time-like and

| ⟨v ×w,v ×w⟩ | = ⟨v,v⟩ ⟨w,w⟩ − (⟨v,w⟩)2.

• det(v,w,v × w) > 0, where det denotes the determinantof 3× 3-matrix consists of three column vectors.

Next, we consider the four dimensional case:

Definition 2.14. For u, v, w ∈ R41, we define

(2.5) u ∧ v ∧w :=

MTH.B405; Sect. 2 (20190721) 22

t(∣∣∣∣∣u1 v1 w1

u2 v2 w2

u3 v3 w3

∣∣∣∣∣ ,∣∣∣∣∣u0 v0 w0

u1 v1 w1

u3 v3 w3

∣∣∣∣∣ ,−∣∣∣∣∣u0 v0 w0

u2 v2 w2

u3 v3 w3

∣∣∣∣∣ ,∣∣∣∣∣u0 v0 w0

u1 v1 w1

u2 v2 w2

∣∣∣∣∣

)

u = t(u0, u1, u2, u3),

v = t(v0, v1, v2, v3),

w = t(w0, w1, w2, w3)

.

In particular, when u is a time-like unit vector and both v andw are perpendicular to u, we write

(2.6) v ×u w := u ∧ v ∧w.

Lemma 2.15. Let u be a time-like unit vector R41. Then for

vectors v, w ∈ R41 perpendicular to,

• v ×u w is perpendicular to u, v, and w,

• if both v are w space-like, v ×u w is space-like vectorsatisfying

⟨v ×u w,v ×u w⟩ = ⟨v,v⟩ ⟨w,w⟩ − (⟨v,w⟩)2.

• det(u,v,w,v ×u w) > 0.

Remark: Topology of Vector Spaces. The Euclidean topol-ogy of Rn is the topology induced by the Euclidean distance

(2.7) d(P,Q) =

√⟨−→PQ,

−→PQ⟩,

where ⟨ , ⟩ is the canonical (positive definite) inner product.When the inner product is not positive definite, (2.7) does not

23 (20190721) MTH.B405; Sect. 2

determine a distance function. We introduce a way to definethe canonical topology of vector space without use of distance:Let V be an n-dimensional vector space.

Definition 2.16. The canonical topology of V is the weakesttopology of V such that any linear functions f : V → R is con-tinuous.

The definition here does not depend on inner products. Thenone can show easily that

Lemma 2.17. The canonical topology of Rn coincides with theEuclidean topology.

Exercises

2-1 Let O(n, s) be as in (2.2).

(1) Show that AB ∈ O(n, s) for any A, B ∈ O(n, s).

(2) Show that A−1 ∈ O(n, s) for any A ∈ O(n, s).

2-2 Let M(n,R) be the set of n × n real matrices with real

components, and identify it with Rn2

, and consider O(n, s)in (2.2) as a subset of M(n,R). Set

W := {X ∈ M(n,R) ; tXJ + JX = O} (J = Js,n−s),

where Js,n−s is the matrix defined in (2.1).

(1) Let A(t) (t ∈ (−ε, ε)) be a smooth curve in M(n,R)satisfying A(0) = I, A(t) ∈ O(n, s). Prove that X :=A′(0) ∈W , where ′ = d/dt.

MTH.B405; Sect. 2 (20190721) 24

(2) For each X ∈W , show that I − tX is invertible for asufficiently small t ∈ R, where I is the n× n identitymatrix.

(3) Set A(t) := (I + t2X)(I − t

2X)−1 for each X ∈ Wand t ∈ R. Prove that

A(t) ∈ O(n, s), A′(0) = X.

2-3 (1) Find an explicit expression of O(2, s) (s = 0, 1).

(2) Determine O(2, 0) ∩O(2, 1).

2-4 (1) Let v =t(v0, v1) is a light-like vector of R2

1. Find alight-like vector w ∈ R2

1 satisfying ⟨v,w⟩ = 1.

(2) Let V be an n-dimensional Minkowski vector spaceandW a space-like subspace of dimension n−2. Showthat there exists two light-like vectors v, w ∈ W⊥

satisfying ⟨v,w⟩ = 1.

(3) In the situation above, show that V =W ⊕Rv⊕Rw.

2-5 Verify Lemma 2.17.

25 (20190721) MTH.B405; Sect. 3

3 Pseudo Euclidean Spaces

Pseudo Euclidean spaces as affine spaces. An affine spaceover a vector space V is a pair of a set A and a map

A×A ∋ (P,Q) 7−→ −→PQ ∈ V

satisfying

(A-1)−→PQ = −−→

QP for all P, Q ∈ A,

(A-2)−→PQ+

−→QR =

−→PR, and

(A-3) for each v ∈ V and P ∈ A, there exists a unique Q ∈ Asatisfying

−→PQ = v.

By (A-3), an affine space over V is identified with A as aset.

Lemma 3.1. Let A be an affine space over a vector space V

and fix a point O ∈ A. Then a map A ∋ Q 7→ −→OQ ∈ V is a

bijection.

Since the point O in Lemma 3.1, A does not have the pre-given origin.

Example 3.2. Let V := Rn and consider a map

Rn × Rn ∋ (P,Q) 7→ −→PQ =

t(q1 − p1, . . . , qn − pn) ∈ Rn

23. April, 2019. Revised: 07. May, 2019

MTH.B405; Sect. 3 (20190721) 26

(P =

t(p1, . . . , pn),Q =

t(q1, . . . , qn)

).

Then Rn can be considered as an affine space over the vectorspace Rn.

Euclidean spaces. Let V be an n-dimensional vector spacewith a positive definite inner product ⟨ , ⟩, andA an affine spaceover V . Then the map A×A → R defined by

(3.1) d(P,Q) :=

√⟨−→PQ,

−→PQ⟩

satisfies the axiom of distance onA. In particular, the Euclideaninner product of Rn induces the Euclidean distance, which de-fines a topology of Rn, called the Euclidean topology.

Verify Lemma 2.17.

Lemma 3.3 (Lemma 2.17, Problem 2-5). The Euclidean topol-ogy of Rn is the weakest topology in which any linear functionon Rn are continuous.

Proof. Since a polynomial on Rn is continuous, so is a linearfunction.

Conversely, take a topology of Rn such that all linear func-tions are continuous. Then, for each j = 1, . . . , n, the coor-dinate function rj : (x1, . . . , xn) 7→ xj is continuous. Hence{(x1, . . . , xn) | a < xj < b} = r−1

j

((a, b)

)is an open set in Rn.

Hence, by taking intersection of such sets, direct products ofopen intervals

(a1, b1)× · · · × (an, bn) ⊂ Rn

27 (20190721) MTH.B405; Sect. 3

are all open. Since the products of intervals generate the Eu-clidean topology, this shows that the Euclidean topology is asubset (as a system of open sets) of the given topology.

Pseudo Euclidean spaces. The affine space based on thepseudo Euclidean vector space Rn

s is called the pseudo Euclideanspace, and denoted by the same symbol Rn

s as its vector space.Since Rn

s can be identified with Rn as affine space, the topologyas in Lemma 3.3 defines a topology of Rn

s . With respect to thistopology, Rn

s is an n-manifold, and the tangent space TPRns at

P ∈ Rns can be identified with Rn

s . This manifold is called thepseudo Euclidean space.

Definition 3.4. A transformation f of Rns (as an affine space)

is said to be an isometry if

⟨−−−−−−→f(P)f(Q),

−−−−−−→f(P)f(Q)

⟩=⟨−→PQ,

−→PQ⟩

holds for each P, Q ∈ Rns , where ⟨ , ⟩ is the canonical inner

product of Rns .

Example 3.5. For a matrix A ∈ O(n, s) and a vector p ∈ Rns ,

the map f : Rns → Rn

s defined by

(3.2) f(x) = Ax+ p(x =

t(x1, . . . , xn) ∈ Rn

s

)

is an isometry.

Theorem 3.6. An isometry of Rns is in the form as (3.2) in

Example 3.5.

MTH.B405; Sect. 3 (20190721) 28

Proof. Let g(x) := f(x) − f(0). Then g is an isometry sat-isfying g(0) = 0. So, g : Rn

s → Rns can be interpreted as an

isometry of the pseudo Euclidean vector space, and hence byProposition 2.2, g is a linear map, that is there exists a matrixA ∈ O(n, s) such that g(x) = Ax. Hence f(x) = Ax+ p wherep = f(0).

The Minkowski vector spaces. In this section, we considerRn+1

1 , the Minkowski vector space. As usual, we write an ele-ment x ∈ Rn+1

1 as

x =t(x0, x1, . . . , xn),

that is, the first component corresponding negative index isnumbered as “0”. We identify the subspace

{(0, x1, x2, . . . , xn) |x1, . . . , xn ∈ R} ⊂ Rn+11

with the n-dimensional Euclidean space Rn. Then each x ∈Rn+1

1 is expressed as

(3.3) x = x0e0 + x,

{e0 =

t(1, 0, . . . , 0), x0 ∈ R,

x = (0, x1, . . . , xn) ∈ Rn.

Using this decomposition, we can write

⟨x,y⟩ = −x0y0 + ⟨x, y⟩ ,

where, y = y0e0+ y is the decomposition of y as (3.3), and ⟨ , ⟩on the right-hand side is the restriction of the inner product ofRn+1

1 to Rn, which is the Euclidean inner product of Rn.

29 (20190721) MTH.B405; Sect. 3

Vectors in Rn+11 are classified in the following 3-causal char-

acters (cf. Definition 2.10):

• v ∈ Rn+11 is said to be space-like if either v = 0 or ⟨v,v⟩ >

0,

• v ∈ Rn+11 is said to be time-like if ⟨v,v⟩ < 0, and

• v ∈ Rn+11 is said to be null or light-like if v = 0 and

⟨v,v⟩ = 0 hold.

Definition 3.7. For a space-like vector v ∈ Rn+11 ,

|v| :=√⟨v,v⟩

is called the norm of v. A space-like vector v is said to be aspace-like unit vector if |v| = 1. On the other hand, a vectorv ∈ Rn+1

1 is called a time-like unit vector if ⟨v,v⟩ = −1 holds.

Proposition 3.8 (cf. Theorem 2.9). For a non-zero vector v ∈Rn+1

1 , the orthogonal complement

v⊥ := {w ∈ Rn+11 | ⟨v,w⟩ = 0}

is

1. an n-dimensional linear subspace of the vector space Rn+11 ,

2. an n-dimensional space-like subspace in Rn+11 when v is

time-like, and

3. an n-dimensional Minkowski subspace, that is a subspaceof Rn+1

1 on which the restriction of the inner product ⟨ , ⟩has the signature (n− 1, 1), if v is space-like.

MTH.B405; Sect. 3 (20190721) 30

4. If v is light-like, v⊥ is an n-dimensional linear subspaceof Rn+1

1 containing v, and all elements in v⊥ \ Rv arespace-like, that is, the restriction of the inner product ofRn+1

1 on v⊥ is a positive semi-definite.

Proof. The linear map L : Rn+11 ∋ w 7→ ⟨w,v⟩ ∈ R is of rank

1 because the inner product is non-degenerate. Hence v⊥ =KerL is an n-dimensional subspace. The assertions 2 and 3 areconclusions of Theorem 2.9. The assertion 4 is proved as follows(Problem 2-4): Decompose v = v0e0 + v, w = w0e0 + w asin (3.3). If both v and w are light-like and perpendicular eachother,

−v20 + |v|2 = 0, −w20 + |w|2 = 0, −v0w0 + ⟨v, w⟩ = 0

hold, and v and w are linearly dependent.

The Lorentz-Minkowski space. The pseudo Euclidean spaceRn+1

1 (as an affine space) is called the Lorentz-Minkowski spaceor Minkowski space of dimension n+1. The following terms areoriginate from the special relativity :

Definition 3.9. For each point P ∈ Rn+11 ,

• the set

ΛP := {Q ∈ Rn+11 | −→PQ is light-like} ∪ {P}

is called the light-cone at P.

31 (20190721) MTH.B405; Sect. 3

• The sets

CP := {Q ∈ Rn+1 | ⟨v,v⟩ ≦ 0 (v =−→PQ)},

C+P := {Q ∈ Rn+1 | ⟨v,v⟩ ≦ 0, v0 > 0 (v =

−→PQ)},

C−P := {Q ∈ Rn+1 | ⟨v,v⟩ ≦ 0, v0 < 0 (v =

−→PQ)}

are called the causal set, the future and the past of P,respectively.

• An isometry f(x) = Ax + p (A ∈ O(n + 1, 1)) of Rn+11

is called a Lorentz transformation. In particular, whenA ∈ SO(n + 1, 1) (resp. A ∈ SO+(n + 1, 1)), it is saidto be orientation preserving (resp. orientation preservingand isochronous or proper).

Proposition 3.10. A Lorentz transformation f(x) = Ax + p(A ∈ O(n + 1, 1)) maps the light-cone (resp. the causal set) atP to the light-cone (resp. the causal set) at f(P). When A ∈SO+(n + 1, 1), it maps the future (resp. the past) at P to thefuture (resp. the past) at f(P)

Remarks on Special Relativity. The Lorentz-Minkowskispace is the mathematical representation of the space-time inthe relativity.

We regard our world as a 3-dimensional Euclidean space,and interpret a point (x0, x1, x2, x3) ∈ R4

1 as the “the point(x1, x2, x3) in the space at the time x0”. In this context, we callthe set R4

1 the space-time. For a mathematical simplicity, wechoose the unit system such that the speed c of the light in the

MTH.B405; Sect. 3 (20190721) 32

vacuum is 1. Denote by γ(x0) = (x1(x0), x2(x0), x3(x0)) ∈ R3

the position of the moving point in the space-time at time x0.Then the parametrized curve

(x0, γ(x0)

)∈ R4

1 is called the worldline of this moving point. In particular, a point passing through(p1, p2, p3) at time 0 with constant velocity and speed 1 (thelight-speed) is one of the generator lines of the light-cone ΛP atP = (0, p1, p2, p3).

An inertial frame of the space-time is a choice of the coor-dinate system (x0, x1, x2, x3) of the space-time.

The fundamental principle of the special relativity are:

(R1) the light-speed are common in all inertial frames, and

(R2) all physical laws are invariant under the change of inertialframes.

A change of the inertial frame is a certain transformationf : R4

1 → R41. The invariance of light-speed (R1) requires that

this transformation maps light-cones to light-cones. It can beshown that such transformations are Lorentz transformationsunder a suitable assumptions. Hence by the principle of rela-tivity (R2) is rephrased as all physical laws are invariant underthe Lorentz transformations.

In this frame work, the words future, past and causal areused in their proper meaning.

For example, consider a line joining two distinct points P, Q

R41 as a motion in the space-time. Decompose

−→PQ as in (3.3):

−→PQ = p0e0 + p.

33 (20190721) MTH.B405; Sect. 3

Then the velocity vector of the motion in R3 ⊂ R41 is

v :=p

p0.

In particular,

• −→PQ is space-like if and only if |v| > 1. In this case, to travelfrom P to Q in the space time with constant velocity, thespeed must be greater than 1 (the light-speed).

• −→PQ is time-like if and only if |v| < 1, which is equivalentto that Q ∈ CP, that is, Q is causal to P. If Q is in thefuture C+

P at P, one can travel from P to Q with constantvelocity and the speed less than 1 (the light-speed). Onthe other hand, if Q is in the past C−

P if P, one can travelfrom Q to P with constant velocity and the speed less than1.

• Q lies in the light-cone ΛP at P if and only if |v| = 1. Ifthis is the case, the line PQ corresponding a motion withconstant velocity and speed 1. In other words, the light-cone ΛP is foliated by the world lines of lights emanatingat P.

Example 3.11. A transformation

f : R21 ∋

(x0x1

)7−→

(y0y1

)=

(cosh θ sinh θsinh θ cosh θ

)(x0x1

)∈ R2

1

of R21 is a proper Lorentz transformation for each θ ∈ R. Un-

der this transformation, each line la : x1 = a in the (x0, x1)-coordinate system is mapped to a line la : y0 = (coth θ)y1 −

MTH.B405; Sect. 3 (20190721) 34

(sech θ)a in the (y0, y1)-coordinate system. Though the line lacorresponding to the point staying at the position x1 = a sta-tionally in the (x0, x1)-coordinate system, the corresponding linein the (y0, y1)-coordinate system represents a motion with speedtanh θ.

Submanifolds and hypersurfaces. In this lecture, mani-folds are assumed as C∞-differentiable manifold. We identifythe pseudo Euclidean space Rn

s with the Euclidean space Rn asa differentiable manifold.

Proposition 3.12 (The implicit function theorem). For a C∞-map F : Rn ⊃ U → Rp defined on a domain U in Rn, assume

M := F−1(0) = {P ∈ Rn |F (P) = 0}is a non-empty set. If the Jacobian matrix dF (P) of F at P is ofrank p for each P ∈M ,M is an (n−p)-dimensional submanifoldof Rn.

In particular, for the case p = 1, we have

Proposition 3.13. Let F : Rn ⊃ U → R be a C∞ functiondefined on a domain U in Rn, and assume

∂F

∂x1(P) = 0

holds at P ∈ F−1(0). Then there exist a neighborhood V ofP = (p1, . . . , pn) ∈ Rn, a neighborhood V ′ of P′ = (p2, . . . , pn) ∈Rn−1 and a C∞-function f : V ′ → R satisfying

M ∩ V = {(f(x2, . . . , xn), x2, . . . , xn) | (x2, . . . , xn) ∈ V ′} .

35 (20190721) MTH.B405; Sect. 3

In other words, M is represented as a graph x1 = f(x2, . . . , xn)on a neighborhood of P.

Under the situation of Proposition 3.13, (x2, . . . , xn) can beconsidered as a local coordinate system ofM on a neighborhoodof P. Hence we have

Corollary 3.14. For a C∞-function F : Rn ⊃ U → R definedon a domain U in Rn, assume

M := F−1(0) = {P ∈ Rn |F (P) = 0}is a non-empty set. If

(dF )P = (Fx1(P), . . . , Fxn

(P)) = 0

holds for each point P ∈ M , M is an (n − 1)-dimensional sub-manifold (i.e. a hypersurface) of Rn.

Example 3.15. A subset

{(x, y, z) ∈ R3 | ax+ by + cz + d = 0} ⊂ R3

is a plane, where a, b, c, d are constants satisfying (a, b, c) =(0, 0, 0).

Example 3.16. The zeros F−1(0) of a quadratic polynomialF (x, y, z) in variables (x, y, z) is called a quadric in R3 if it isa smooth surface in R3. Ellipsoids, hyperbolids of one sheet,hyperboloids of two sheets, elliptic parabolas and hyperbolicparabolas are quadrics1.

1For explicit expressions of these surfaces, see, for example, Section 6of “Differential Geometry of Curves and Surfaces” by M. Umehara and K.Yamada, World Scieintific.

MTH.B405; Sect. 3 (20190721) 36

Exercises

3-1 Prove 4 in Proposition 3.8.

3-2 Prove Propositoin3.10.

3-3 Show that O(n, s) is a smooth submanifold in M(n,R) =Rn2

. (Hint: O(n, s) = F−1(O), where F (A) = tAJA− J ,J = Js,n−s).

37 (20190721) MTH.B405; Sect. 4

4 Non-degenerate submanifolds.

Immersions. In this section, a differentiable manifold or sim-ply a manifold means a C∞-differentiable manifold, and abbre-viate C∞-differentiable as differentiable.

Definition 4.1. A differentiable map

f : M → N

between differentiable manifolds M and N is an immersion ifthe differential map

(df)P : TPM −→ Tf(P)N

of f is injective at each point P ∈ M . Here, TPM and Tf(P)Ndenote the tangent space of M and N at points P and f(P),respectively.

The differential (df)P of the map f as in Definition 4.1 is alinear map defined as

(df)P(v) : C∞(N) ∋ φ 7−→ v(φ ◦ f) ∈ R

for v ∈ TPM .

07. May, 2019. Revised: 14. May, 2018

MTH.B405; Sect. 4 (20190721) 38

Lemma 4.2. Let P ∈ M be a point of a manifold M , andv ∈ TPM . Take a curve γ on M satisfying

γ : (−ε, ε) →M, γ(0) = P,dγ

dt(0) = v.

Then for a differentiable map f : M → N , it holds that

(df)P(v) =d

dt

∣∣∣∣t=0

f(γ(t)

).

Remark 4.3 (The matrix representatives of differentials).Let f : M → N be a differentiable map and take local coordinatesystems (x1, . . . , xm) and (y1, . . . , yn) around P ∈M and f(P) ∈N , respectively, and let Df(P) be the matrix representative of(df)P with respect to the bases

{(∂

∂x1

)

P

, . . . ,

(∂

∂xm

)

P

},

{(∂

∂y1

)

f(P)

, . . . ,

(∂

∂yn

)

f(P)

}

of TPM and Tf(P)N , respectively, that is, Df(P) is an n ×m-matrix satisfying

[(df)P

(∂

∂x1

), . . . , (df)P

(∂

∂xm

)]

=

[(∂

∂y1

), . . . ,

(∂

∂yn

)]Df(P).

39 (20190721) MTH.B405; Sect. 4

Then it coincides with the Jacobian matrix

Df(P) :=

(∂f1∂x1

). . .

(∂f1∂xm

)

.... . .

...(∂fn∂x1

). . .

(∂fn∂xm

)

of f at P, where, fj ’s are components of f(x1, . . . , xm) withrespect to the coordinate system (yj).

In particular, when N = Rn, Df is expressed as(∂f

∂x1, . . . ,

∂f

∂xm

)= (fx1 , . . . , fxm),

where each fxj(j = 1, . . . ,m) is the partial derivative of f with

respect to the variable xj as a (column) vector-valued function.

Corollary 4.4. A map f : M → Rn is an immersion if and onlyif fx1(P), . . . , fxm(P) are linearly independent at each P ∈M ,where m = dimM . Here, (x1, . . . , xm) is a local coordinatesystem of M around P.

A subset M ⊂ Rn is called a submanifold of Rn if thereexists a structure of manifold onM such that the inclusion mapι : M → Rn is an immersion. In particular, when dimM =n− 1, the submanifold M is called a hypersurface. In this case,the tangent space TPM is considered as a linear subspace ofRn(= TPRn).

Lemma 4.5. The derivative of the inclusion map ι : M → Rn

of a submanifold M ⊂ Rn satisfies

dιP(v) = v (v ∈ TPM).

MTH.B405; Sect. 4 (20190721) 40

Proof. For an arbitrary v ∈ TPM , take a curve γ(t) on M suchthat γ(0) = P, γ(0) = v, where ˙ = d/dt. Then ι ◦ γ(t) = γ(t)holds as a curve in Rn, Lemma 4.2 yields

dιP(v) =d

dt

∣∣∣∣t=0

ι ◦ γ(t) = γ(0) = v ∈ TPRn.

Pseudo Riemannian manifolds.

Definition 4.6. A (pseudo) Riemannian metric on a manifoldM is a collection of inner products

gP : TPM × TPM → R

for all P ∈M such that

(4.1) M ∋ P 7−→ gP(XP, YP) ∈ R

is a C∞-function on M for arbitrary smooth vector fields

X : M ∋ P 7→ XP ∈ TPM, Y : M ∋ P 7→ YP ∈ TPM.

The function (4.1) is denoted by g(X,Y ), or often written as⟨ , ⟩. A pair (M, g) of a manifoldM and a (pseudo) Riemannianmetric g is called a (pseudo) Riemannian manifold.

Lemma 4.7. Let M be a connected manifold and let g be apseudo Riemannian metric. Then the map

M ∋ P 7→ the signature of gP

is a constant map.

41 (20190721) MTH.B405; Sect. 4

Definition 4.8. Let (M, g) be a connected pseudo Riemannianmanifold. The signature of (M, g) is the signature (n+, n−) ofgP, which does not depend on P.

Example 4.9. Rns is a pseudo Riemannian manifold of signa-

ture (n− s, s).

Induced metric. In this section, we consider geometry ofsubmanifolds in the pseudo Euclidean space Rn

s .Let f : M → Rn

s be an immersion. Then

dfP(TPM) ⊂ Tf(P)Rns = Rn

s

is a linear subspace of dimension m := dimM . The inducedmetric on M by f is a collection of symmetric bilinear forms⟨ , ⟩P on TPM (P ∈M) defined as

(4.2) ⟨v,w⟩P := ⟨dfP(v), dfP(w)⟩ v,w ∈ TPM.

In particular, when M ⊂ Rns is a submanifold, then the induced

metric is the restriction of the inner product ⟨ , ⟩ on TPM ⊂ Rns .

Definition 4.10. Let f : M → Rns be an immersion. A point

P is said to be

• non-degenerate point if the induced metric ⟨ , ⟩P is non-degenerate,

• space-like point if the induced metric ⟨ , ⟩P is positive def-inite, and

MTH.B405; Sect. 4 (20190721) 42

• degenerate point if the induced metric ⟨ , ⟩P is a degener-ate bilinear form.

In addition, an immersion f is said to be

• non-degenerate if all P ∈ M are non-degenerate points,and

• space-like if all P ∈M are space-like points.

Example 4.11. LetM ⊂ Rns be a non-degenerate (resp. space-

like) submanifold. Then (M, ⟨ , ⟩) is a pseudo-Riemannian (resp.Riemannian) manifold, where ⟨ , ⟩ is the induced metric.

Hypersurfaces and normal vectors. As a special case, weconsider hypersurfaces of Rn+1

s .

Definition 4.12. For a smooth function F : Rn+1s → R and

P ∈ Rn+1s , we define

gradF (P) := Jt,n+1−t

∂F

∂x0(P)

∂F

∂x1(P)

...

∂F

∂xn(P)

=

(−Is OO In+1−s

)

∂F

∂x0(P)

∂F

∂x1(P)

...

∂F

∂xn(P)

,

which is called the gradient vector of F at P, where (x0, x1, . . . , xn)is the coordinate system of Rn+1

s .

43 (20190721) MTH.B405; Sect. 4

The gradient vector gradF (P) does not vanish at P if andonly if dF (P) = 0. Hence Corollary 3.14 can be rewritten as

Corollary 4.13. Let F : U → R be a smooth function on a do-main U of Rn+1

s such that M := F−1({0}) = ∅. If gradF (P) =0 holds on each point P ∈ M , M is a smooth hypersurface ofRn+1

s .

Proposition 4.14. Assume a function F : Rn+1s → R satisfies

the assumptions of Corollary 4.13. Then the tangent space ofM = F−1({0}) at P ∈M is given by

TPM =(gradF (P)

)⊥.

Proof. Let v ∈ TPM . Then there exists a curve

γ : (−ε, ε) ∋ t 7−→ γ(t) ∈M ⊂ Rn+1s (ε > 0)

on M such that γ(0) = P, γ(0) = v (˙ = d/dt). Since γ(t) ∈Mfor all t, φ(t) := F (γ(t)) is identically zero. On the other hand,

if we write γ(t) =t(x0(t), . . . , xn(t)

),

γ(t) =t(x0(t), . . . , xn(t)

).

Hence

0 = φ(0) =d

dt

∣∣∣∣t=0

F(γ(t)

)=

n∑

j=0

∂F

∂xj

(γ(0)

) dxjdt

∣∣∣∣t=0

=

n∑

j=0

∂F

∂xj(P)

dxjdt

∣∣∣∣t=0

= ⟨gradF (P), γ(0)⟩ = ⟨gradF (P),v⟩ .

MTH.B405; Sect. 4 (20190721) 44

Therefore, v ∈(gradF (P)

)⊥, that is, TPM ⊂

(gradF (P)

)⊥.

Moreover, since TPM is a tangent space of n-dimensional mani-fold M , it is a vector space of dimension n. On the other hand,

by (1) of Proposition 3.8,(gradF (P)

)⊥is of dimension n. Thus

we have the conclusion.

In particular, Proposition 3.8 yields

Proposition 4.15. Assume a function F : Rn+11 → R on the

Lorentz-Minkowski space Rn+11 satisfies the assumption of Corol-

lary 4.13. Then P ∈M = F−1({0}) is• a space-like point on M if and only if gradF (P) is time-like,

• a time-like point, that is, the induced metric is indefi-nite (i.e., Minkowski, in this case) on M if and only ifgradF (P) is space-like, and

• degenerate point ofM if and only if gradF (P) is light-like.

Definition 4.16. Let M ⊂ Rn+1s be a non-degenerate hyper-

surface. A vector ν(P) ∈ Rn+1s = TPRn+1

s perpendicular toTPM and satisfies | ⟨ν(P), ν(P)⟩ | = 1 is called the unit normalvector of M at P. The map P 7→ ν(P) is called the unit normalvector field of M .

Proposition 4.14 yields

Proposition 4.17. The unit normal vector of a non-degeneratehypersurface M := F−1(0) ⊂ Rn+1

s is given by

ν(P) = ± gradF (P)

| ⟨gradF (P), gradF (P)⟩ |1/2

45 (20190721) MTH.B405; Sect. 4

where F is a smooth function on Rn+1s such that gradF does

not vanish on M .

Examples: Let c be a constant, and define a function qc : Rn+1s →

R by

(4.3) qc(x) := ⟨x,x⟩ − c.

Here we identify the point P ∈ Rn+1s with the position vector

x =−→OP. We set

(4.4) Qc := q−1c ({0}) = {x ∈ Rn+1

s | ⟨x,x⟩ = c}.

The case s = 0:, that is, the case of hypersurfaces in the Eu-clidean space Rn+1,

• when c > 0, Qc is a hypersphere centered at the originwith radius

√c, in particular, when n = 2, Qc is a sphere

of radius√c.

• when c = 0, Q0 consists only of the origin, and

• when c < 0, Qc = ∅.

The case s = 1:, that is, the case of hypersurfaces in Lorentz-Minkowski space Rn+1

1 . In this case, Qc = ∅ for an arbitrary c.Since

(4.5) grad qc(x) = 2x,

it vanishes if and only if x = 0.

MTH.B405; Sect. 4 (20190721) 46

• Since 0 ∈ Qc when c = 0, Qc is a submanifold in Rn+11 .

• When c = 0, Qc is not a submanifold in Rn+11 on a neigh-

borhood of the origin. For P ∈ Qc \ {0}, there exists aneighborhood U ⊂ Rn+1

1 of P such that Qc ∩ U is a sub-manifold of Rn+1

1 .

47 (20190721) MTH.B405; Sect. 4

Exercises

4-1 Let Qc ⊂ Rn+11 be the subset of the Lorentz-Minkowski

space Rn+11 defined as in (4.4).

(1) Find the value of c such that Qc is a degenerate hypersurface.

(2) Find the condition for c for Qc to be a space-like(time-like) hypersurface.

(3) Draw the figure of Q0, Q1, Q−1 when n = 2.

4-2 Recall the vector product of R31 Definition 2.12:

v ×w :=t(

−∣∣∣∣v1 w1

v2 w2

∣∣∣∣ ,∣∣∣∣v2 w2

v0 w0

∣∣∣∣ ,∣∣∣∣v0 w0

v1 w1

∣∣∣∣)

for v =t(v0, v1, v2), w =

t(w0, w1, w2) ∈ R3

1. Let f : M →R3

1 be an immersion of 2-dimensional manifold M intothe Lorentz-Minkowski 3-space. Take a local coordinatesystem (u, v) in M .

(1) Prove that f is a space-like (resp. time-like) surfaceif and only if fu × fv is time-like (resp. space-like).

(2) Let g(x1, x2) be a function of two variables definedon a domain U ⊂ R2, and consider an immersion

f(x1, x2) :=(g(x1, x2), x1, x2

)

from U into R31, whose image is the graph of g. Find

a condition for such a graph to be space-like (resp.time-like, degenerate).

MTH.B405; Sect. 4 (20190721) 48

4-3 Show that

M := {x ∈ Rn+12 | ⟨x,x⟩ = 1}

is a non-degenerate hypersurface in Rn+12 , and compute

its signature.

49 (20190721) MTH.B405; Sect. 5

5 Geodesics

Revised: 21. May, 2019

Orthogonal decomposition of the tangent space. LetM ⊂ Rn+1

s be a non-degenerate submanifold2. By non-degeneracy,the restriction of the inner product of Rn+1

s to each tangentspace TPM is non-degenerate. Then by Theorem 2.9, the or-thogonal decomposition

(5.1) Rn+1s = TPRn+1

s = TPM ⊕NP,(NP := (TPM)⊥

)

holds for each P ∈ M . The restriction of the inner product⟨ , ⟩ of Rn+1

s to NP is non-degenerate. We call NP the normalspace of M at P. According to this decomposition, each vectorv ∈ TPRn+1

s = Rn+1s can be decomposed uniquely as

(5.2) v = [v]T+ [v]

N[v]

T ∈ TPM, [v]N ∈ NP.

We call [v]T(resp. [v]

N) the tangential part (resp. normal part)

of v.For the case that M is a hypersurface of Rn+1

s (that is,dimM = n), the normal space is spanned by the unit normalvector, that is, NP = RνP, where νP is the unit normal vectorof M at P:

(5.3) Rn+1s = TPRn+1

s = TPM ⊕ RνP.

14. May, 2019.2Not necessarily of codimension one at this moment.

MTH.B405; Sect. 5 (20190721) 50

In this case,

(5.4) [v]N= ε ⟨v, νP⟩ νP, [v]

T= v − [v]

N

hold, where ε = ⟨νP, νP⟩ ∈ {−1, 1}.

Vector fields along curves. A curve on a non-degeneratesubmanifold M of Rn+1

s is a map

γ : J ∋ t 7−→ γ(t) ∈M ⊂ Rn+1s

where J ⊂ R is an interval. A curve γ is said to be of class Cr

if γ is a Cr-map as a map into Rn+1s . The following fact is a

direct conclusion of the definition of differentiability of maps:

Lemma 5.1. A map γ : J → M ⊂ Rn+1s is of class Cr as a

map into a differentiable manifold M if and only if it is of classCr as a map into Rn+1

s .

From now on, by a word smooth, we mean of class C∞. Forexample, a smooth curve γ means a curve γ of class C∞.

Definition 5.2. Let γ : J → M ⊂ Rn+1s be a smooth curve on

M . A smooth vector field on M along γ is a map

X : J ∋ t 7−→ X(t) ∈ Tγ(t)M ⊂ Rn+1s

which is of class C∞ as a map from J to Rn+1s .

Example 5.3. Let γ : J → M ⊂ Rn+1s be a smooth curve.

Then

γ : J ∋ t 7−→ γ(t) =dγ

dt(t) ∈ Tγ(t)M

51 (20190721) MTH.B405; Sect. 5

is a smooth vector field along γ, called the velocity vector fieldof the curve γ.

Definition 5.4. A smooth curve γ : J →M is said to be regularif γ(t) = 0 for all t ∈ J . A regular curve γ is called non-degenerate if ⟨γ(t), γ(t)⟩ = 0, that is, γ(t) is not a light-likevector, for each t ∈ J . When ⟨γ, γ⟩ > 0 (resp. ⟨γ, γ⟩ < 0), it issaid to be space-like (resp. time-like).

Example 5.5. Consider

Q1 := {x ∈ R41 | ⟨x,x⟩ = 1}

as defined in (4.4) for n = 3 and s = 1. Then

γ1(t) :=(sinh t, 0, 0, cosh t

),

γ2(t) :=(0, cos t, sin t, 0

),

γ3(t) :=(t, 1, 0, t

)

are regular curves on Q1, which are time-like, space-like anddegenerate, respectively.

Lemma 5.6. Let γ : J →M be a non-degenerate regular curveon a non-degenerate submanifold M ⊂ Rn+1

s . Then there existsa parameter change t = t(τ) such that

∣∣⟨γ′, γ′⟩∣∣ = 1, where γ′(τ) =

dγ

dτ

(t(τ)

).

Proof. Set

τ(t) :=

∫ t

t0

√| ⟨γ(u), γ(u)⟩ | du.

MTH.B405; Sect. 5 (20190721) 52

Since γ is non-degenerate, dτ/dt =√

| ⟨γ, γ⟩ | > 0. Then thereexists the inverse function t = t(τ) of τ(t), and the chain ruleyields the conclusion.

We call the parameter τ as in Lemma 5.6 the arc-lengthparameter of the curve. The arc-length parameter of a time-like curve in a Lorentzian manifold M is often called the propertime.

Covariant derivative of vector fields along curves. Letγ be a smooth regular curve on a non-degenerate submanifoldM ⊂ Rn+1

s , and X a smooth vector field of M along γ3. Thenwe obtain a map

X : J ∋ t 7−→ X(t) =dX

dt∈ Rn+1

s ,

which is not a vector field on M , in general.

Definition 5.7. The vector field

∇dtX(t) = ∇γ(t)X(t) :=

[X(t)

]T∈ Tγ(t)M

of M along γ is called the covariant derivative of X along γ,where [∗]T denotes the tangential component as in (5.2).

Definition 5.8. The covariant derivative

(5.5)∇dtγ(t) = ∇γ(t)γ(t) := [γ(t)]

T ∈ Tγ(t)M

3From now on, we assume all objects are of class C∞ and omit the wordsmooth.

53 (20190721) MTH.B405; Sect. 5

of γ along γ is called the acceleration of the curve γ.

Lemma 5.9. For each curve γ on M , it holds that

d

dt⟨γ(t), γ(t)⟩ = 2

⟨∇γ(t)γ(t), γ(t)

⟩.

Proof. Since γ(t) ∈ Tγ(t)M ,

d

dt⟨γ(t), γ(t)⟩ = 2 ⟨γ(t), γ(t)⟩ = 2

⟨[γ(t)]

T+ [γ(t)]

N, γ(t)

⟩

= 2⟨[γ(t)]

T, γ(t)

⟩

= 2⟨∇γ(t)γ(t), γ(t)

⟩.

Geodesics and pre-geodesics.

Definition 5.10. A regular curve γ on a non-degenerate sub-manifold M ⊂ Rn+1

s is called a pre-geodesic if ∇γ γ is propor-tional to γ, and called a geodesic if ∇γ γ = 0 holds.

Lemma 5.11. Let γ be a pre-geodesic on a non-degenerate sub-manifold M ⊂ Rn+1

s . Then there exists a parameter changet = t(τ) such that γ(τ) := γ

(t(τ)

)is a geodesic.

Proof. Problem 5-1.

Lemma 5.12. Let γ be a geodesic of a non-degenerate subman-ifold M ⊂ Rn+1

s . Then ⟨γ(t), γ(t)⟩ does not depend on t.

Proof. A direct conclusion of Lemma 5.9.

MTH.B405; Sect. 5 (20190721) 54

Local expressions. Here, we give an expression of geodesicsin the local coordinate system. Let M ⊂ Rn+1

s be a non-degenerate submanifold and take a local coordinate neighbor-hood (U ;u1, . . . , um) of M , where m = dimM4. Then the in-clusion map ι : M → Rn+1

s induces an immersion

(5.6) f : U ∋ (u1, . . . , um) 7−→ f(u1, . . . , um) ∈M ⊂ Rn+1s ,

here we identify the coordinate neighborhood U ⊂ M with aregion of Rm. We call such an f a (local) parametrization ofM .Set

(5.7) gij :=

⟨∂f

∂ui,∂f

∂uj

⟩(= gji

)(i, j = 1, . . . ,m),

which is a component of the induced metric g := ⟨ , ⟩ |TPMwwith respect to the basis

{(∂

∂u1

)

P

, . . . ,

(∂

∂um

)

P

}

of TPM for each P ∈ U . Since the induced metric is non-degenerate, the m×m-matrix (gij) is a regular matrix at eachpoint P. We denote by (gij) the inverse matrix of (gij):

(5.8)

m∑

k=1

gikgkj = δji =

{1 (i = j)

0 (i = j)

4Here we adopt the classical notation of indices, that is, we use thesuperscript as uj , instead of uj . In this context, we distinguish superscriptsand subscripts.

55 (20190721) MTH.B405; Sect. 5

Lemma 5.13. Let γ is a curve in U ⊂M and express

γ(t) = f(u1(t), . . . , um(t)

),

where f : U → M is a local parametrization of M as in (5.6).Then

γ =m∑

j=1

duj

dt

∂f

∂uj(5.9)

∇γ γ =

m∑

j=1

d

2uj

dt2+

m∑

k,l=1

Γ jkl

duk

dt

dul

dt

∂f

∂uj(5.10)

hold, where

(5.11) Γ kij =

1

2

m∑

l=1

gkl(∂gil∂uj

+∂glj∂ui

− ∂gij∂ul

).

Proof. Problem 5-2.

The functions Γ kij of (5.11) are called the Christoffel symbols

with respect to the local coordinate system (u1, . . . , um).

Theorem 5.14. Let M ⊂ Rn+1s be a non-degenerate subman-

ifold and fix P ∈ M . Then for each v ∈ TPM , there exists aunique geodesic γv : (−ε, ε) →M satisfying

γv(0) = P, γv(0) = v.

MTH.B405; Sect. 5 (20190721) 56

Proof. Take a local coordinate system (U ;u1, . . . , um) ofM aroundP. Then a curve γ(t) = f(u1(t), . . . , un(t)) (in the parametriza-tion as in (5.6)) is geodesic if and only if

(5.12)d2uj

dt2+

m∑

k,l=1

Γ jkl

duk

dt

dul

dt= 0 (j = 1, . . . ,m)

because of(5.10). Let (u10, . . . , un0 ) be the coordinates of the

point P and

v = v1(

∂

∂u1

)

P

+ · · ·+ vn(

∂

∂un

)

P

.

Then the initial condition γ(0) = P, γ(0) = v corresponds to

(5.13) uj(0) = uj0,duj

dt(0) = vj .

Since the Christoffel symbols Γ kij are functions in (u1, . . . , un),

(5.12) is a normal form of an ordinary differential equation ofsecond order with respect to the unkowns uj(t) (j = 1, . . . ,m).Hence by the fundamental theorem for ordinary differential equa-tions, we have the unique solution of (5.12) under the initialcondition (5.13).

Definition 5.15. A geodesic γ on M is said to be complete ifthe domain of definition of γ(t) is (can be extended to) R.

Hopf Rinow’s theorem. In this subsection, we consider aconnected Riemannian manifolds, that is, the metrics are as-sumed to be positive definite. For a curve γ : J → M in a

57 (20190721) MTH.B405; Sect. 5

Riemannian manifold M , we define the length of γ by

(5.14) L(γ) :=∫ b

a

⟨γ(t), γ(t)⟩1/2 dt(≧ 0).

We denote by CP,Q the set of piecewise smooth curves joiningpoints P and Q in M , and define

(5.15) d(P,Q) := infγ∈CP,Q

L(γ).

Since M is assumed to be connected, the function d : M ×M →R is well-defined. We can prove the following, which can befound in textbooks on Riemannian geometry:

Fact 5.16. Let d be a function defined in (5.15). Then

• d is a distance function on M ,

• the topology on M induced by the distance d coincides withthe topology of M as a manifold.

Moreover, if the length L(γ) of the curve γ : J →M is equal tod(γ(a), γ(b)), γ is a pre-geodesic.

We call the function d the distance function on M inducedfrom the Riemannian metric.

Fact 5.17 (The Hopf-Rinow theorem). Let M be a connectedRiemannian manifold. Then the following conditions are equiv-alent:

• All geodesics in M are complete.

MTH.B405; Sect. 5 (20190721) 58

• All geodesics starting at a point P ∈M are complete.

• The distance function d induced from the Riemannian met-ric is complete (as a distance function).

• All divergent path on M5have infinite lengths.

• All bounded set on M with respect to d are relatively com-pact.

Moreover, if these properties are satisfied, then, for each P, Q ∈M , there exists a geodesic joining P and Q ∈ M , whose lengthis d(P,Q).

Examples

Example 5.18 ((Pseudo) Euclidean spaces). Consider Rns be

the submanifold of itself. Taking the canonical coordinate sys-tem (x1, . . . , xn), the coefficients gij in (5.7) are constants. Henceall the Christoffel symbols vanish, and (5.12) turns to be d2uj/dt2 =0 (j = 1, . . . , n). Hence a curve (u1(t), . . . , un(t)) is a geodesicif and only if all uj(t)’s are a linear functions in t, that is, ageodesic is expressed as

γ(t) = tv + p,

that is, a straight line with constant velocity v. Since this isdefined on R, Rn

s is complete. In particular, when s = 0, the

5A curve γ : [0,∞) → M is called a divergent path if for an arbitrarycompact set K in M , there exists a number tK such that γ([tk,∞)) ⊂M \K.

59 (20190721) MTH.B405; Sect. 5

distance d(P,Q) is the length of the line segment joining P andQ.

Example 5.19 (The spheres.). Consider Sn = Sn(1) ⊂ Rn+1

in the Euclidean (n+1)-space, which coincides with Q1 in (4.4)for s = 0. Let p ∈ Sn and identify it with its position vector,and take a unit vector v ∈ TpS

n. Set

γv(t) = (cos t)p+ (sin t)v.

Then we have

• ⟨γv(t), γv(t)⟩ = 1, that is, γv(t) is a curve on Sn,

• γv(t) = −γ(t), that is, γv(t) is perpendicular to the tan-gent space of Sn at γv(t),

• γv(0) = p, γv(0) = v, and

• the image γv(R) of γv is the intersection of Sn and theplane spanned by {p,v} passing through the origin.

A circle obtained as the intersection of the sphere and a planepassing through the origin is called the great circle. Since allgeodesics are defined on R, Sn is a complete Riemannian man-ifold.

Example 5.20 (The hyperbolic spaces.). Consider Q−1 as in(4.4) for s = 1. Since this is disconnected, we write a connectedcomponent of it by

Hn = {x =t(x0, . . . , xn) ∈ Rn+1

1 ; ⟨x,x⟩ = −1, x0 > 0}.

MTH.B405; Sect. 5 (20190721) 60

Let p ∈ Hn and take a unit vector v ∈ TpHn. We set

γv(t) = (cosh t)p+ (sinh t)v.

Then

• ⟨γv(t), γv(t)⟩ = −1, that is, γv(t) is a curve on Hn,

• γv(t) = γv(t), that is, γv(t) is perpendicular to the tan-gent space of Hn at γv(t),

• γv(0) = p, γv(0) = v,

• the image γv(R) is the intersection of Hn and the planespanned by {p,v} passing through the origin.

Exercises

5-1 Prove Lemma 5.11.

5-2 Prove Lemma 5.13.

5-3 Verify Example 5.20.

61 (20190721) MTH.B405; Sect. 6

6 Geodesics and Christoffel symbols

Local Expressions (continued). Let M ⊂ Rn+1s be a non-

degenerate submanifold and take a local coordinate neighbor-hood (U ;u1, . . . , um) of M , where m = dimM . Then the inclu-sion map ι : M → Rn+1

s induces an immersion of U ⊂ Rm intoRn+1

s , called a local parametrization of M , as

(6.1) f : U ∋ (u1, . . . , um) 7→ f(u1, . . . , um) ∈M ⊂ Rn+1s .

The components of the induced metric g := ⟨ , ⟩ |TM

(6.2) gij :=

⟨∂f

∂ui,∂f

∂uj

⟩(= gji

)(i, j = 1, . . . ,m),

with respect to the basis{(

∂

∂u1

)

P

, . . . ,

(∂

∂um

)

P

}

of TPM for each P ∈ U induces a smooth matrix valued function((g)) : U → Sym(m,R), where Sym(m,R) is the set of m ×m-symmetric matrices of real components. Since the inducedmetric is non-degenerate, the matrix ((g)) := (gij) is a regularmatrix at each point P. Each component gij of the inversematrix (gij) := ((g))−1 is a smooth function of U .

Definition 6.1. Under the situation above, the smooth func-tions

Γ kij :=

1

2

m∑

l=1

gkl(∂gil∂uj

+∂glj∂ui

− ∂gij∂ul

)(i, j, k = 1, . . . ,m)

21. May, 2019. Revised: 28. May, 2019

MTH.B405; Sect. 6 (20190721) 62

on U are called the Christoffel symbol with respect to the localcoordinate system (u1, . . . , um).

Lemma 6.2. The Christoffel symbols satisfy

Γ kij = Γ k

ji,∂gij∂uk

=m∑

l=1

(gjlΓ

lik + gilΓ

lkj

)

for i, j, k = 1, . . . ,m.

Proof. By symmetricity of the metric, gkl = glk holds for eachk, l. Then

Γ kji =

1

2

m∑

l=1

gkl(∂gjl∂ui

+∂gli∂uj

− ∂gji∂ul

)

=1

2

m∑

l=1

gkl(∂glj∂ui

+∂gil∂uj

− ∂gij∂ul

)= Γ k

ij .

On the other hand, since

m∑

l=1

gjlΓlik =

m∑

l=1

gjl1

2

m∑

s=1

gls(∂gis∂uk

+∂gsk∂ui

− ∂gik∂us

)

=1

2

m∑

s=1

m∑

l=1

(gjlgls)

(∂gis∂uk

+∂gsk∂ui

− ∂gik∂us

)

=1

2

m∑

s=1

δsj

(∂gis∂uk

+∂gsk∂ui

− ∂gik∂us

)

=1

2

(∂gij∂uk

+∂gjk∂ui

− ∂gik∂uj

),

63 (20190721) MTH.B405; Sect. 6

m∑

l=1

gilΓlkj =

m∑

l=1

gilΓljk

=1

2

(∂gij∂uk

+∂gik∂uj

− ∂gjk∂ui

),

the second equality follows.

Lemma 6.3. The local parametrization f as in (6.1) satisfies

(6.3)

[∂2f

∂ui∂uj

]T=

m∑

k=1

Γ kij

∂f

∂uk

for i, j = 1, . . . ,m.

Proof. Since {∂f

∂u1(P), . . . ,

∂f

∂um(P)

}

is a basis of TPM for each P ∈ U , there exist smooth functions∆k

ij such that

(6.4)

[∂2f

∂ui∂uj

]T=

m∑

k=1

∆kij

∂f

∂uk.

Taking the inner product with ∂f/∂ul, (6.4) turns to be

(6.5)

⟨[∂2f

∂ui∂uj

]T,∂f

∂ul

⟩=

m∑

k=1

gkl∆kij .

MTH.B405; Sect. 6 (20190721) 64

The left-hand side of (6.5) computed as⟨[

∂2f

∂ui∂uj

]T,∂f

∂ul

⟩=

⟨∂2f

∂ui∂uj,∂f

∂ul

⟩

=∂

∂ui

⟨∂f

∂uj,∂f

∂ul

⟩−⟨∂f

∂uj,∂2f

∂ui∂ul

⟩

=∂gjl∂ui

−⟨∂f

∂uj,∂2f

∂ul∂ui

⟩

=∂gjl∂ui

− ∂

∂ul

⟨∂f

∂uj,∂f

∂ui

⟩+

⟨∂2f

∂ul∂uj,∂f

l∂ui

⟩

=∂gjl∂ui

− ∂gij∂ul

+

⟨∂2f

∂uj∂ul,∂f

∂ui

⟩

=∂gjl∂ui

− ∂gij∂ul

+∂

∂uj

⟨∂f

∂ul,∂f

∂ui

⟩−⟨∂f

∂ul,∂2f

∂uj∂ui

⟩

=∂gjl∂ui

− ∂gij∂ul

+∂gil∂uj

−⟨

∂2f

∂ui∂uj,∂f

∂ul

⟩

=∂gjl∂ui

− ∂gij∂ul

+∂gil∂uj

−⟨[

∂2f

∂ui∂uj

]T,∂f

∂ul

⟩,

and hence⟨[∂2f

∂ui∂uj

]T,∂f

∂ul

⟩=

1

2

(∂gjl∂ui

− ∂gij∂ul

+∂gil∂uj

),

here, we used that the Leibnitz rule

d

dt⟨a(t), b(t)⟩ =

⟨da

dt(t), b(t)

⟩+

⟨a(t),

db

dt(t)

⟩

65 (20190721) MTH.B405; Sect. 6

for vector-valued functions a(t), b(t) and the (pseudo) Euclideaninner product ⟨ , ⟩. Hence (6.5) yields

m∑

k=1

gkl∆kij =

1

2

(∂gil∂uj

+∂glj∂ui

− ∂gij∂ul

),

and then

Γ sij =

1

2

m∑

l=1

gsl(∂gil∂uj

+∂glj∂ui

− ∂gij∂ul

)

=

m∑

k=1

m∑

l=1

gslgkl∆kij =

m∑

k=1

δsk∆kij = ∆s

ij ,

which yields the conclusion.

Using Lemma 6.3 and the chain-rule, we have

Lemma 6.4 (cf. Lemma 5.13). Let γ is a curve in U ⊂M andexpress

γ(t) = f(u1(t), . . . , um(t)

),

where f : U → M is a local parametrization of M as in (6.1).Then

γ =

m∑

j=1

duj

dt

∂f

∂uj(6.6)

∇γ γ =m∑

j=1

d

2uj

dt2+

m∑

k,l=1

Γ jkl

duk

dt

dul

dt

∂f

∂uj.(6.7)

MTH.B405; Sect. 6 (20190721) 66

Theorem 6.5 (cf. Theorem 5.14). Let M ⊂ Rn+1s be a non-

degenerate submanifold and fix P ∈ M . Then for each v ∈TPM , there exists a unique geodesic γv : (−ε, ε) →M satisfying

γv(0) = P, γv(0) = v.

Proof. Take a local coordinate system (U ;u1, . . . , um) ofM aroundP. Then a curve γ(t) = f(u1(t), . . . , um(t)) (in the parametriza-tion as in (6.1)) is geodesic if and only if

(6.8)d2uj

dt2+

m∑

k,l=1

Γ jkl

duk

dt

dul

dt= 0 (j = 1, . . . ,m)

because of (6.7). Let (u10, . . . , um0 ) be the coordinates of the

point P and

v = v1(

∂

∂u1

)

P

+ · · ·+ vm(

∂

∂um

)

P

.

Then the initial condition γ(0) = P, γ(0) = v corresponds to

(6.9) uj(0) = uj0,duj

dt(0) = vj .

Since the Christoffel symbols Γ kij are functions in (u1, . . . , um),

(6.8) is a normal form of an ordinary differential equation ofsecond order with respect to the unkonwns uj(t) (j = 1, . . . ,m).Hence by the fundamental theorem for ordinary differential equa-tions, we have the unique solution of (6.8) under the initial con-dition (6.9).

Definition 6.6. A geodesic γ on M is said to be complete ifthe domain of definition of γ(t) is (can be extended to) R.

67 (20190721) MTH.B405; Sect. 6

The parameter change and the Christoffel symbols. Let(U ;u1, . . . , um) and (U ; u1, . . . , um) be two local coordinate sys-tems of M around P ∈M , and denote by

f = f(u1, . . . , um), f = f(u1, . . . , um)

the local parametrization of M with respect to the coordinatesystem (u1, . . . , um) and (u1, . . . , um), respectively. Then theJacobian matrix

J :=

∂u1

∂u1 . . . ∂u1

∂um

.... . .

...∂um

∂u1 . . . ∂um

∂um

is a regular matrix at each point P ∈ U ∩ U . The following isthe immediate consequence of the chain-rule:

Lemma 6.7. For each point P ∈ U ∩ U , then

∂f

∂ua=

m∑

j=1

∂uj

∂ua∂f

∂uj,

that is,

(∂f

∂u1, . . . ,

∂f

∂um

)=

(∂f

∂u1, . . . ,

∂f

∂um

)J

holds at P.

Lemma 6.8. Let v ∈ TPM and set

v =

m∑

j=1

vj(∂f

∂uj

)

P

=

m∑

a=1

va

(∂f

∂ua

)

P

.

MTH.B405; Sect. 6 (20190721) 68

Then it holds that

vj =

m∑

a=1

∂uj

∂uava, that is,

v1

...vm

= J

v1

...vm

.

Denote by gij and gab be the components of the inducedmetric with respect to the coordinate system (uj) and (ua),respectively.

Lemma 6.9. It holds that

gab =m∑

i,j=1

∂ui

∂ua∂uj

∂ubgij , that is, ((g)) = tJ((g))J,

where ((g)) = (gij) and ((g)) = (gab) are the symmetric matricesconsisting of the components of the metric.

Finally, we denote by Γ kij and Γ c

ab the Christoffel symbols

with respect to the local coordinate systems (uj) and (ua), re-spectively.

Lemma 6.10. It holds that

m∑

c=1

Γ cab

∂uk

∂uc=

m∑

i,j=1

(∂ui

∂ua∂uj

∂ubΓ kij

)+

∂2uk

∂ua∂ub.

Definition 6.11. A (pseudo) Riemannian manifold (M, g) issaid to be flat if, for all P ∈ M , there exists a local coordinatesystem around P such that all Christoffel symbols with respectto the coordinate system vanish identically.

69 (20190721) MTH.B405; Sect. 6

The Hopf-Rinow theorem (restated). In this subsection,we consider a connected Riemannian manifolds, that is, themetrics are assumed to be positive definite. For a curve γ : [a, b] →M on the Riemannian manifold M , we define the length of γ by

(6.10) L(γ) :=∫ b

a

⟨γ(t), γ(t)⟩1/2 dt(≧ 0).

We denote by CP,Q the set of piecewise smooth curves joiningpoints P and Q in M , and define

(6.11) d(P,Q) := infγ∈CP,Q

L(γ).

Since M is assumed to be connected, the function d : M ×M →R is well-defined. We can prove the following, which can befound in textbooks on Riemannian geometry:

Fact 6.12. Let d be a function defined in (6.11). Then

• d is a distance function on M ,

• the topology on M induced by the distance d coincides withthe topology of M as a manifold.

Moreover, if the length L(γ) of the curve γ : J →M is equal tod(γ(a), γ(b)), γ is a pre-geodesic.

We call the function d the distance function on M inducedfrom the Riemannian metric.

Fact 6.13 (The Hopf-Rinow theorem). Let M be a connectedRiemannian manifold. Then the following assertions are equiv-alent.

MTH.B405; Sect. 6 (20190721) 70

• All geodesics in M are complete.

• All geodesics starting at a point P ∈M are complete.

• The distance function d induced from the Riemannian met-ric is complete (as a distance function).

• All divergent path on M have infinite lengths6.

• All bounded set on M with respect to d are relatively com-pact.

Moreover, if these properties are satisfied, then, for each P, Q ∈M , there exists a geodesic joining P and Q ∈ M , whose lengthis d(P,Q).

Exercises

6-1H (Beltrami’s pseudosphere) Let U := {(u1, u2) ∈ S1 ×R ; u2 > 0} and set M := f(U) ⊂ R3, where S1 ={(cos t, sin t) ∈ R2 ; t ∈ R} and

f(u1, u2) :=(sechu2 cosu1, sechu2 sinu1, u2 − tanhu2

).

Take a geodesic γ(t) := f(u1(t), u2(t)) on M with theinitial condition

u1(0) = a andd

dtu2(0) = 0.

6A curve γ : [0,∞) → M is called a divergent path if for an arbitrarycompact set K in M , there exists a number tK such that γ([tk,∞)) ⊂M \K.

71 (20190721) MTH.B405; Sect. 6

Prove that there exists a constant b such that

(u1(t)− a

)2+ cosh2 u2(t) = b.

(Hint: There exists a constant α such that u1 = α cosh2 u2).

6-2H (Jacobi’s first integral of the geodesic flow on ellipsoids.)Let M ⊂ R3 be an ellipsoid

M :=

{(x, y, z) ∈ R3 ;

x2

a2+y2

b2+z2

c2= 1

},

and let γ(t) =(x(t), y(t), z(t)

)be a geodesic on M with

γ′(t) =(u(t), v(t), w(t)

). Prove that

J(t) :=

(x(t)2

a4+y(t)2

b4+z(t)2

c4

)

×(u(t)2

a2+v(t)2

b2+w(t)2

c2

)

is constant in t. (Hint: Let NP =(

xa2 ,

yb2 ,

zc2

). Then

N is (not necessarily unit) normal vector of M at P =(x, y, z). By definition, γ satisfies γ′′(t) = λ(t)N(t) whereλ(t) is a real valued function, N(t) := Nγ(t) and

′ = d/dt.Taking the inner product with N(t), we have that λ =⟨γ′′, N⟩ / ⟨N,N⟩ = −⟨γ′, N ′⟩ / ⟨N,N⟩.)

6-3H Let U := {(u1, u2) ; u1 > 0} ⊂ R2 and

f : U ∋ (u1, u2) 7→ (u1 cosu2, u1 sinu2) ∈ R2,

MTH.B405; Sect. 6 (20190721) 72

which gives a local parametrization of f(U). Find thecoordinate change (u1, u2) 7→ (u1, u2) such that all the

Christoffel symbols Γ cab with respect to (u1, u2) vanish

identically.

73 (20190721) MTH.B405; Sect. 7

7 Final Remarks

A review: pre-geodesics and geodesics. Recall that acurve γ(t) on a (pseudo) Riemannian manifold (M, g) is saidto be a geodesic (resp. pregeodesic) if

∇γ γ = 0(resp. ∇γ γ//γ

).

Proposition 7.1. If a curve γ is a geodesic on (M, g), a curveγ obtained by a parameter change of γ is a pre-geodesic. Con-versely, assume a pregeodesic γ is non-degenerate. Then thereparametrization γ of γ with the arclength parameter is a geodesic.

Proof. We give a proof for a submanifold of a pseudo Euclideanspace Rn+1

s . Let γ(τ) be a geodesic and t = t(τ) a parameterchange. Then we have

γ′ =dt

dτγ and γ′′ =

d2t

dτ2γ +

(dt

dτ

)2

γ,

where ′ = d/dτ and ˙ = d/dt. Hence

∇γ′ γ′ = [γ′′]T=d2t

dτ2γ +

(dt

dτ

)2

[γ]T=d2t

dτ2γ

yields the first assertion.On the other hand, let γ(τ) be a pre-geodesic, and take the

arc-length parameter t of γ(t) := γ(τ(t))). Since γ(t) has theunit-length γ is perpendicular to γ. Here γ is parallel to γ

28. May, 2019. 22. June, 2019

MTH.B405; Sect. 7 (20190721) 74

because γ is a pregeodesic, and the notion of pregeodesic doesnot depend on parameter changes. Hence ∇γ γ = [γ]

Tvanishes

identically, namely, γ is a geodesic.

Beltrami’s pseudosphere. Recall Problem 6-1: We let

(7.1) f(u, v) : U = {(u, v) ∈ R2 ; v > 0} ∋ (u, v)

7→ (sech v cosu, sech v sinu, v − tanh v) ∈ R3.

Then f is an immersion of the upper-half plane U into theEuclidean 3-space R3. Since f is 2π-periodic in the variableu, it can be considered as an immersion of S1 × R+, whereS1 = R/(2πZ) is the unit circle. In fact, the image of f is a sur-face of revolution with respect to the z-axis whose profile curveγ(v) = (sech v, 0, v − tanh v) on the xz-plane7.

Since

fu = sech v(− sinu, cosu, 0) and

fv = tanh v(− sech v cosu,− sech v sinu, tanh v),

the components gij of the induced metric are expressed as

g11 = sech2 v, g12 = g21 = 0, g22 = tanh2 v.

Hence the Christoffel symbol (cf. Definition 6.1) is computed as

Γ 111 = Γ 1

22 = Γ 212 = 0,

7This profile curve is called the tractrix, and the surface is named Bel-trami’s pseudosphere. See page 260 of [UY] for figures of the pseudosphere.

75 (20190721) MTH.B405; Sect. 7

Γ 112 = Γ 1

21 = − tanh v,

Γ 211 = Γ 2

22 = sech v csch v.

Then by (6.8), we have that a curve γ(t) = (u(t), v(t)) is ageodesic on the pseudosphere if and only if it satisfies

(7.2) u− 2uv tanh v = 0, v + csch v sech v(u2 + v2) = 0,

where “dot” denotes the derivative with respect to t.

Proposition 7.2. Let γ(t) = (u(t), v(t)) be a curve on theupper-half plane U which gives a geodesic on the pseudosphere(7.1), and satisfy

(7.3) u(0) = a,d

dtv(0) = 0,

where a ∈ R is a constant. Then there exists a constant c suchthat (

u(t)− a)2

+ cosh2 v(t) = c2

for all t.

Proof. By the first equation,

(7.4)u(t)

u(t)= 2

d

dt

(log cosh v(t)

), and then u(t) = A cosh2 v(t)

holds for some constant A.Set

φ(t) :=(u(t)− a

)2+ cosh2 v(t).

MTH.B405; Sect. 7 (20190721) 76

Then by (7.4), we have

φ = 2u(u− a) + 2v cosh v sinh v

= 2(A cosh2 v(u− a) + v cosh v sinh v

)

= 2 cosh2 v(A(u− a) + v tanh v

)= 2 cosh2 vψ(t),

here, we setψ(t) := A(u− a) + v tanh v.

Differentiating this, we have

ψ = Au+ v tanh v + (v)2 sech2 v

= A2 cosh2 v − tanh v csch v sech v(u2 + v2)

= A2 cosh2 v − sech2 v(A2 cosh4 v + v2) + (v)2 sech2 v = 0.

Hence, by (7.3), we have ψ(t) = const = ψ(0) = 0, and thenφ(t) = 0. Therefore φ(t) = c2 is a constant.

Theorem 7.3. A curve γ(t) = (u(t), v(t)) is a pregeodesic onthe pseudosphere if and only if

u(t) = a, or(u(t)− a

)2+ cosh2 v(t) = c2

for some constants a and c.

Proof. Let P = (u0, v0) (v0 > 0) and take a direction w =(w1, w2) ∈ TPU . If w1 = 0, it is easy to verify that the curveγ(t) = (u0, t) is a pregeodesic. On the other hand, if w1 = 0,there exists a constant a satisfying

w1(u0 − a) + w2 cosh v0 sinh v0 = 0.

77 (20190721) MTH.B405; Sect. 7

Then the curve {(u, v) ; (u−a)2+cosh v2 = c2} (c2 := (u0−a)2+cosh2 v0) is a pregeodesic, by virtue of Proposition 7.2. Since ageodesic is uniquely determined by the initial point and initialvelocity, all pre geodesics are obtained in this manner.

The Hyperbolic Plane. Consider a parameter change

(u, v) 7→ (u,w), w = cosh v

on the parametrization of the pseudosphere in (7.1). Then itwill be expressed as f : W → R3, where

f(u,w) :=

(1

wcosu,

1

wsinu, cosh−1 w − 1/

√w2 − 1

)

and W := {(u,w) ; w > 1}. The components gij of the inducedmetric with respect to the coordinate system (u,w) are

g11 = g22 =1

w2, g12 = 0.

which are extended to the upper-half plane W = {(u,w) ; w >0}, and the geodesics are expressed as

u = constant, or (u− a)2 + w2 = c2.

Thus we have the Riemannian manifold (W , g), and one caneasily show that it is complete, which is called the hyperbolicplane, and denoted by H2.

The hyperbolic plane is a model of the non-Euclidean geome-try, found by Bolyai Janos, Nikolai Ivanovich Lobachevsky, and(possibly) Carl Friedlich Gauss in 19th century, independently.

MTH.B405; Sect. 7 (20190721) 78

Though the hyperbolic plane obtained by extending the pseu-dosphere, which is a surface in R3, the following fact is known:

Fact 7.4 (David Hilbert, 1901). There exists no isometric im-mersion of the hyperbolic plane into the Euclidean 3-space.

On the other hand,

Theorem 7.5. A quadric

M := {x = (x0, x1, x2) ∈ R31 ; ⟨x,x⟩ = −1, x0 > 0}

in the Mikowski 3-space is isometric to the hyperbolic plane.

Proof. Set

φ : M ∋ (x0, x1, x2) 7−→ ξ + i η =x1 + ix2

1 + x0∈ C (i =

√−1).

Then φ is a diffeomorphism from M to D := {z ∈ C ; |z| < 1},that is, (ξ, η) 7→ φ−1(ξ+i η) gives a parametrization of M . Thecomponents gij of the induced metric is computed as

g11 = g22 =4

(1 + ξ2 + η2)2, g12 = 0.

On the other hand, the diffeomorphism

ψ : U = {z ∈ C ; Im z > 0} ∋ z 7→ z + 1

z − 1∈ D

yields the parametrization φ−1◦ψ : U →M and the components(gij) of the induced metric can be computed as

g11 = g22 =1

(Im z)2, g12 = 0.

79 (20190721) MTH.B405; Sect. 7

References

[UY] M. Umehara and K. Yamada, Differential Geometryof Curves and Surfaces, World Scientific, 2015.