B.

Local Coordinates,

Curvature,

and Area

I CHP a surface

I : V → I a coordinate chart,

V c IN'

for any I E 1mF we saw in section HI that

Tp -2 = Im DIE where I EV s # Itg )ip

It we use coordinates luv ) on Vc IR'

then Im Dtp is spanned by

Fiel and 3747

we denote these by

Tip ) and Ttp)

recall we THE

a::÷÷. : '

(9) by Iv

The 1st fundamental form of I C IR'

on Tp -2 is

Iip ) = Tp -2 x Tp -2 → RU

⇐, Tra ) 1-7 F. WT

Iers in IR

'

Tdot product in 473

so I Ip ) is an inner product on Tp -2

if we set a Cpt = I Cpt . I Ip )

b CF) = it Lp ) . Tip )

c Cp) = ftp.vlp )

then I Cpt can be represented by the matrix

a Cp) b C p )

( bird cops)

to see this note any vectors I, I C- Tp -2 can be written

I = x,

a- txzv = I# in the basis it

,I

I -

- y ,I tyzv = [ Y

,:]now I CptII Ty ) = I . I

= ( x ,I exit ) . ( y ,

I t yet )

= x, y ,

a t X, Yztxzy ,) b t X z Yz C

-

- Exist lab:) I

so in the basis a,

I

Iim -

- I 997, %:3)and this measures lengths of vectors and anglesbetween vectors in Tp -2

now for g- in V define

gig) : 1122×1122 → IR

by gig) I wi . %)= I HTT) ) ( Dfg Cut )

, Dfg Cod )

= LDF-quiDo ( Dig CE ) ]

so we can represent g in the basis Fu,Zo by

a. tan: . it¥:DIdea is g represents I in local coordinates

I : V → E

we callg a Riemannian metric ( but for Ic IR

'it is essentially

equivalent to the 1st fundamental form,

more on this

later )

many computations on I can be done in local coordinates

using g Icu)

example . I-€7i. I

1-1 F -- to I

a b

length (pi ) = fall f' Hill It

=Sab IlfEt ) d t

=SabIttDDzIDDz⇐I 'HD dt

= Jab

g.CI#)lEHI.I'HTdt--baHI'ttNggdtmeans length of

veto r using g

Recall from calc HI :

if a surface I in IRS is parameterized byI : V → IR

3

Vc 1122

( u, v )

and R -

- I ( v ) a -2

then the area of R is

Area C R ) = fullTill dado

where it = Futit = Iot

note : HaxTH = KUTIHEH sin 0 ¥,%= kill HEH 1-6507

-

- thinkinf-fEi=

HuTTuvTP_µ=ac-bT

= deft = detg

so Area IR ) = Svidetgdudr

example : 5 c 1123 be the unit sphere

I : y → 5 : @ioTi-sfu.v,Fut)

( I

{ Cair ) E IR ? u 't v'

c I }

so a- Eat -

- toe If ] to ] -1¥)

similarly of =/

"

a -- ii. a = It 7¥ -

- IET

b = T.ci = Url- U2 - ✓

2

C = T . I =

I - U2 - V2

and

gun , = IECIiitlet U -

- upper hemisphere of 53 so

area ( U ) = £,

idetg du do

= f,

,(FyzH- wth - rt - u 're)hedv= S

, ,

Cl - on - v)"

Z

dado

= J,

deed v

change →

to polar = J, ,

rdr do

word - s

= Sis ! door

= ZIT f - if ) to = 2T

Recall to compute Gauss and mean curvature we can take

the normal vector it to the surface and consider

the shape operator

Sp CE ) -- - to 457

then the Gauss curvature is Kip ) = def Sp

and the mean curvature is Hip ) = I tr Sp} " G " 5

now if I :X → E gives local coordinates on -2 then

Tp -2 is spanned by it = (Dfg ) ⇐ ) where Feit =p

ri -- DIE ) Br )

and

I =u

the XIII

in the basis Tiv for Tp E we can write Sp as

Sp-

- Laa!,

9

a;) just as we did above

before lemma 3

we need to find aij

Set A = Split . I

B = Sp tu ) . I

C-= Sp IT ) . I

Remark : The 2nd fundamental form of E atp is

I Cpt :Tp Ex Tp -2 → IR

④ , WI) t Sp I Tv,

) . I

so in the basis air we can represent Itp ) as

( AB Be ) just like we did for the

1st fundamental form above

Spi and II Ipt have the same information,

so I only mention

this since some books prefer I and some S

note : A = Sp tu ) . I = I ai ,T t

az ,I ) . I = a

, ,a tar ,

b

B = Spca ) . I = ( a, ,

I taut ) . F = a. ,b t q ,

c

Sp I T ) .I = ( aizu t ant ) . I = a. za t an b

C = Sp ( I ) . I = ( aizu tazz I) . I = a, z

b t an C

this is equivalent to saying

*:3 .

. la:" a:X ::3or Ca:: 'Il ::S

"

= .tl:71 ⇐is ]so we have proven the followinglemma 6 :

with the above notation

Ktp ) = def Sp = deftlydef Iip )

= ACa c - b -

and

Hip ) = Etr Sp= Iz Ac-2bBt#

ac - 62

Remark : In the proof of lemma 2 we saw that

(E) . a- =- T . Faa

Wolf -

-

- t.FI ,

City ) . -v= - t . too

where Fascist = LD ( DIE I Eu )),

# ) ] for g- st.

I toil=p

this can be very helpful when computingexample a > r so fixed constants

I I u.

v ) = ( ( atr cos u ) cosy ( a tr cos u ) sin v,

r sin u )

ZIT

i.torus ( donut )

vi. Ia = ( - rsinucosv,

- r sinus inv,

roosa )

of = If = flatroosa ) sin v,

Latrcosu ) cosy o )

SO : a = I. I = r2

b = vi. T = O

c = catrosa ,a

9 -

- ⑨be )

and

N- = Y¥#r,=

. . .

= - ( cos u cost, cos a Shiv

,Smu )

w

exercise

and

Faa = I - rcosucosv,

- rcosusinv,

- r since )

Ffa = ( r sinus inv,

- rsinucosv ,O )

Izz = ( - I at rosa ) loser,

- ( at rios a) Sisu,

o )

SO :

A = Stu ) . it = F.

Faa= r

B -

- Stu ) . I = I .

Taz = O

C -

- Shh . T = F. try = ( at rcosu ) cos u

Has :

kcu.ir ) det I=

AC - B'

=rcatrcosu ) cos u

- -def I ac - b'

r- ( atr cosa )

2

= Losar( at r cos a )

v Kip )

+ - + →

I 'Iz u

C. Some Implications of Curvature

note Sp is particularly scaiple if K,

= KE C

i.e. Split ) = CT

we call such a point is an um bi tic pointThat .

If every point of E c 1123 is umbilici then

I is a part of a plane if K = O or

I is a part of a sphere of radius it K > o

( need I path connected for this )

note the Gauss curvature at an umbilici point is 20

Proof : let I :all,uj→E c IR

'be a word

.chart

set a = Diner,I Iu )

T =D Far,( ⇒ } span Team,

-2

since all points are umbilici we have

$ might →K I = Sp tu ) = - NIU

depend -, KJ = Spelt ) - - Tv

on IIn the proof of lemma 2.3 we saw

( to) ,

=

,)

,

we actually did this for Ibut same computationworks for T too

SO

Half -

- Nth

Ky I t K Tiz = KIT t K Fa

but recall to - (DIfIuL)z= DID In.

.nl#Dfiu..fZr)j-DlDttu.nlErDfcu.nlFu)

lemma 2.3

computation =

FasoKy I = Kiev

and since I and of are linearly independent we see

Kg = Kui = O

so K is constant on coordinate charts

exercise . Show K constant on all of -2

Hint : pig E I take path I from p IoT

cover link ) by charts and consider

overlap of charts

Case I : K -

- o,

then E part of a plane

Proofs: K -

- O ⇒ Safir ) = O fir and p

so Ff (f) =-

Sp CF ) = o tf E and p

Intuitively,

it not changing so I must be in a-

plane perpendicular to I

Rigorously: recall the equation for a plane perp .

tot is

a. o.o

,

and the plane perp to I but through I is

the set of all I satisfying,

of

f. co - pt = o

for a fixed FEE let D= Tcp )

now for any g- EE,

let I :{ o ,c ] → I be an

arc with I cos =p and ICD= q

Set f- C t ) = ( ICH - p ) . N' I ICH )

f'

It ) = It tho NTI KD t I ICH - p ) . D Faye,

LI ' CED- -

= O since I'M c- Tache - S, #

( I 'It )

= - ( Ict ) - p ) o S, #

l I 'It ) )

= O

so fit ) is constant !

f- Co) = ( F - p ) . Dcp ) = o

i. fu ) = O

I I

(g- - p ) . N' Cg) but It NTI KD = DNF#CI '

HDit

I = - S, #

( Ita )= O

IF-F) ' Iso NTI ) = Bcp) =D

So for any point g- EE is in the plane

(Ep

) . it to

Case 2 : K to

Proof: let I : V → E be a word chart

Consider : Flav) = I luv ) txt NTI Cair )-

.

note ¥ Flair ) = Father ) txt I Diff.

. . ,) I Fat curl )

=

Futian- ¥ Spca .nl Father )

- -

= Zu f - ¥ Eat = O

similarly ZrF=o

so F is constant, say F- Car ) = I

that is,

we have I luv ) t INT Fears) = E

so It Flair ) - Elle It - ¥NTfTuvDH = ¥

that is any point in the image of I is on

the sphere of radius I about I

the Gauss curvature is K = KK

so the radius is

exercise See other word charts for -2 are

on the same sphereL#

That 8 .

If I E IRS is a compact surface,

then there

is some point § E E such that Kip ) > o

7 roof : consider f : I → IR ip-llplf-p.ptf- is continuous

a result from calculus implies that any continuous

function on a compact set has a maximum Cand mint

so I fo E -2 such that f- I p ) E f Cfo ) V p E E

..

siriasis:.

Ideas. at Do,

E curves more ( in all directions ) than

the sphere of radius It pill about the org in

so all principal curvatures are larger than

those of the sphere ⇐1¥ )

so we expect K Cpo )z¥p > 0

Tomakethisrigorous.com pate the curvature of

-2 at PJ in direction I E Up -2

recall,

there is a curve

I :L - E. e ] → E

such that I co ) =

ToI 'co ) = I

HI ' kN =L htt

from lemma I we know← normal to E

K pot) = I "

lol . N' Cfo )

now lets fail it

for any tangent vector E e Tpo -2 let

§ be a path in I set.

§ Lot Fo

§'co ) = I

not : fo B- It ) = B-Lt ) . Jlt )

So dat top It ) I⇐ o

-

- F'

to ) . B- Colt I Co) - Bio)

= 2p-o.to

but top has a maximum at O ( since f- on

-2 does )

so 2 Do . I = ¥ top # o

= O

re.

E is perpendicular to Fo VT E TEEthus D= ftp.T,

is ( or- this

,but sign wont

matter )

now consider fo I = I . I

since

⇐o is a maximum of fo I we see

dat fo I # 0=0

and

d¥ fo Itt= o

£0

OZ htt Ifee.I .IT/o=dzl2I' E) to

= 2K".It I !I ' ) to

= 2 ( I"

to ) . Fo t I )

so PJ . I' '

to ) E - I

and Kpocci ) = I' '

co ) . I = I " Colo PI" Doll

± -

upto < 0

for any I E Up.

-2

so both principal curvatures are a- t

- ltpoll

>I

> o:.

K Ipo ) = Kika- ltpoll

'

#