23
Differential Geometry and its Applications 21 (2004) 229–251 www.elsevier.com/locate/difgeo Pappus type theorems for motions along a submanifold M. Carmen Domingo-Juan a , Vicente Miquel b,,1 a Departamento de Matemáticas Económicoempresariales, Universidad de Valencia, Valencia, Spain b Departamento de Geometría y Topología, Universidad de Valencia, Burjasot, Valencia, Spain Received 31 January 2003; received in revised form 25 September 2003 Communicated by L. Vanhecke Abstract We study the volumes volume(D) of a domain D and volume(C) of a hypersurface C obtained by a motion along a submanifold P of a space form M n λ . We show: (a) volume(D) depends only on the second fundamental form of P , whereas volume(C) depends on all the i th fundamental forms of P , (b) when the domain that we move D 0 has its q -centre of mass on P , volume(D) does not depend on the mean curvature of P , (c) when D 0 is q -symmetric, volume(D) depends only on the intrinsic curvature tensor of P ; and (d) if the image of P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO(n q d), and C is closed, then volume(C) does not depend on the i th fundamental forms of P for i> 2 if and only if the hypersurface that we move is a revolution hypersurface (of the geodesic (n q)-plane orthogonal to P ) around a d -dimensional geodesic plane. 2004 Elsevier B.V. All rights reserved. MSC: 53C20; 53C21; 53C99; 53A07 Keywords: Pappus formulae; Tube; Volume; Space form; Parallel motion; Motion along a submanifold; Comparison theorem 1. Introduction In [14], Weyl gave a formula for the volumes of a tube and a tubular hypersurface around a submanifold P of the Euclidean space and the sphere. First he obtained a formula for the volume which shows it depends only on the second fundamental form of P and the radius of the tube. Then, he applied * Corresponding author. E-mail addresses: [email protected] (M.C. Domingo-Juan), [email protected] (V. Miquel). 1 Work partially supported by a DGI Grant No. BFM2001-3548 and AVCiT GRUPOS 03/169. 0926-2245/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.difgeo.2004.05.005

Pappus type theorems for motions along a submanifold · 2017. 2. 26. · Pappus type theorems for motions along a submanifold M. Carmen Domingo-Juana, Vicente Miquelb,∗,1 a Departamento

  • Upload
    others

  • View
    7

  • Download
    0

Embed Size (px)

Citation preview

  • tale

    orem

    und awhich

    ied

    Differential Geometry and its Applications 21 (2004) 229–251www.elsevier.com/locate/difgeo

    Pappus type theorems for motions along a submanifold

    M. Carmen Domingo-Juana, Vicente Miquelb,∗,1

    a Departamento de Matemáticas Económicoempresariales, Universidad de Valencia, Valencia, Spainb Departamento de Geometría y Topología, Universidad de Valencia, Burjasot, Valencia, Spain

    Received 31 January 2003; received in revised form 25 September 2003

    Communicated by L. Vanhecke

    Abstract

    We study the volumes volume(D) of a domainD and volume(C) of a hypersurfaceC obtained by a motionalong a submanifoldP of a space formMnλ . We show: (a) volume(D) depends only on the second fundamenform of P , whereas volume(C) depends on all theith fundamental forms ofP , (b) when the domain that wmoveD0 has itsq-centre of mass onP , volume(D) does not depend on the mean curvature ofP , (c) whenD0 isq-symmetric, volume(D) depends only on the intrinsic curvature tensor ofP ; and (d) if the image ofP by the ln ofthe motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra ofSO(n − q − d),andC is closed, then volume(C) does not depend on theith fundamental forms ofP for i > 2 if and only if thehypersurface that we move is a revolution hypersurface (of the geodesic(n − q)-plane orthogonal toP ) around ad-dimensional geodesic plane. 2004 Elsevier B.V. All rights reserved.

    MSC:53C20; 53C21; 53C99; 53A07

    Keywords:Pappus formulae; Tube; Volume; Space form; Parallel motion; Motion along a submanifold; Comparison the

    1. Introduction

    In [14], Weyl gave a formula for the volumes of a tube and a tubular hypersurface arosubmanifoldP of the Euclidean space and the sphere. First he obtained a formula for the volumeshows it depends only on the second fundamental form ofP and the radius of the tube. Then, he appl

    * Corresponding author.E-mail addresses:[email protected] (M.C. Domingo-Juan), [email protected] (V. Miquel).

    1 Work partially supported by a DGI Grant No. BFM2001-3548 and AVCiT GRUPOS 03/169.

    0926-2245/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.difgeo.2004.05.005

    http://www.elsevier.com/locate/difgeo

  • 230 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    sic

    deeperoodman

    eng a

    c

    on

    ic

    if weially only

    efinesection

    ld

    f

    asse’s

    invariant theory to show that, in fact, the dependence onP reduces to a dependence on the intringeometry ofP (see[6] for a modern approach and further references).

    In [7], Gray and the second author initiated a way, via Pappus type theorems, to get aunderstanding of these formulae. The starting point was the computations by Goodman and Gin [4] (completed by Pursell and Flanders in[11] and[3]) generalizing Pappus formulae for the volumof a domain (or a surface) inR3 obtained by the motion of a plane domain (or a plane curve) alocurve inR3. In [7], all these formulae were generalized to simply connected space formsMnλ of constantsectional curvatureλ and arbitrary dimensionn. Given a curvec(t) in Mnλ , let P0 be the totally geodesihypersurface ofMnλ through c(0) and orthogonal toc(t), let D0 be a domain ofP0 and letC0 be ahypersurface ofP0, and letD andC be, respectively, the domain and the hypersurface ofMnλ obtainedby a motion alongc(t) of D0 andC0, respectively. In[7] it is shown that:

    (a) volume(D) depends only on the first curvature ofc(t), and not on the otherith curvatures;(b) if the centre of mass ofD0 is on the curve, then volume(D) does not depend on the motion nor

    the curvature ofc(t), only on the length ofc(t) and the geometry ofD0;(c) for parallel motions, it is still true for volume(C) that it depends only on the length ofc(t) and the

    geometry ofC0 whenc(0) is the centre of mass ofC0.

    In [2], Gual and the authors studied volume(C) in more detail, and showed, among others, that,

    (d) for most motions, the above nice property of volume(C) only holds ifC0 is contained in a geodessphere, that is, ifC is very similar to a tubular hypersurface.

    In conclusion: all the nice properties of the Weyl’s formula remain for motions along curvesmove domains with the centre of mass on the curve, and, for hypersurfaces, this happens essentwith parallel motions.

    In this paper we complete this study by considering motions along submanifoldsP of Mnλ of dimensionq � 2, which was the original (and more complicated) setting of Weyl’s tube formula. First, we dwhat we understand by a motion along a submanifold (we will define also tubes of nonsphericalat the end ofSection 2). Then, we shall obtain a formula for volume(D) (Theorem 3.1) whenD is adomain obtained by the motion alongP of a domainDp contained in the totally geodesic submanifoM

    n−qλp of M

    nλ throughp ∈ P of dimensionn−q tangent to the vector spaceNpP normal toP atp. From

    this formula we get:

    (a′) volume(D) depends on the second fundamental form ofP , but not on the successiveithfundamental forms fori > 2;

    (b′.i) if p is theq-centre of mass ofDp, then volume(D) does not depend on the mean curvature oP ,and, ifDp is central symmetric with respect top, then volume(D) does not depend on the(2i+1)thmean curvatures ofP (seeSection 4).

    After this remark, we introduce the concept ofq-symmetry with respect to a pointp (Definition 4.2)of a domainDp ⊂ Mn−qλp . It is an “integral symmetry” which generalizes the concept of centre of m(Dp is 1-symmetric with respect top if and only if p is the centre of mass ofDp). Then we show a largfamily of examples ofq-symmetric domains (Proposition 4.3). Our main result in this context is a Weyl

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 231

    a tubeetry

    e

    m

    type formula forvolume(D) whenDp is q-symmetric with respect top (Theorem 4.4). A consequenceis that

    (b′.ii) when Dp is q-symmetric with respect top, volume(D) depends only onDp and the intrinsicgeometry ofP .

    This completely solves the question of the influence of the ball-like shape of the section ofaround a submanifoldP on the fact that the volume of the tube depends only on the intrinsic geomof P : what really matters is that the sections of the tube beq-symmetric. On the other hand, thnecessity of the hypothesis ofq-symmetry is shown by some simple examples given inSection 4(beforeDefinition 4.2). It is also remarkable thatthe symmetry ofDp must increase with the dimension ofP inorder to have a Weyl’s type formula for the volume.

    We also obtain a formula for volume(C) (Theorem 5.1) whenC is a hypersurface ofMnλ obtained bythe motion alongP of a hypersurfaceCp contained inM

    n−qλp . From this formula we get that

    (c′) in general, volume(C) depends on all theith fundamental forms ofP , but volume(C) is biggerthan some quantity depending on the second fundamental form ofP and the motion. IfCp isq-symmetric, this quantity does not depend on the motion, it depends only onCp and the intrinsicgeometry ofP , and it is an universal lower bound for volume(C) in the set of the motions ofCpalongP .

    (d′) Concerning volume(C), our main results areTheorems 6.4 and 6.6. We show thatfor most(generic)motions(all nonparallel motions ifn − q = 2), the lower bound forvolume(C) is attained only ifCp is contained in a geodesic sphere ofM

    n−qλp with centre atp (Theorem 6.4(a) and Corollary 6.7).

    Among the remaining motions, for most of them the lower bound is attained only ifCp is a revolutionsubmanifold(Theorem 6.4(b)). The precise statement ofTheorem 6.4that we shall see inSection 6has the following intuition: “when the motion ofCp runs over all kind of rotations inSO(n−q −d),if volume(C) attains its lower bound, then there is an action ofSO(n − q − d) overCp with orbitsof codimensiond”. Moreover, inTheorem 6.6we characterize a pair(motionϕ, hypersurfaceCp)on whichvolume(C) attains its lower bound.

    Now, some notation:M will denote an arbitrary Riemannian manifold of dimensionn, andP will be a regular submanifold

    of M of dimensionq.NP will denote the normal bundle ofP in M , andW will be the maximal neighborhood ofP in NP

    on which the exponential map, exp, is a diffeomorphism.D will denote the normal connection onNP induced by the Levi-Civita connection∇ on M .Let p be a fixed point ofP . Ppt will denote theD-parallel transport fromp to α(t) along some

    geodesicα(t) of P satisfyingα(0) = p.For everyx ∈ P , and every unitN ∈ NxP , LN will denote the Weingarten map ofP at x in the

    direction ofN .For everyx ∈ P , τt will denote the∇-parallel transport fromx along some geodesic of the for

    expx tw, with w ∈ NxP and|w| = 1.rx(z) = dist(x, z), where dist is the distance inM , x ∈ P andz ∈ expx(W ∩ NxP ). In this situation,

    the distance tox in M is the same that the distance tox in expx(W ∩NxP ).

  • 232 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    a

    ng ae we

    ss,there).

    �p :P → R will denote the map defined by�p(x) = distP (p, x), where distP is the distance inP .The symbol( ′ ) will denote the usual derivative for a function fromR to R, the tangent vector for

    curve inMnλ , and the covariant derivative for a vector field along a curve.For everyλ ∈ R, sλ :R → R will denote the solution of the equations′′ + λs = 0 with the initial

    conditionss(0) = 0 ands′(0) = 1; andcλ = s′λ.Mnλ will denote a simply connected space form of dimensionn and constant sectional curvatureλ.For anyx ∈ P , Mn−qλx will denote the totally geodesic submanifold throughx, of dimensionn − q,

    tangent toNxP . For eachy ∈ Mn−qλx , Ny(x) will denote the unit vector atx tangent to the minimizinggeodesic fromx to y.

    Given any subsetF of a hypersurfaceCp of Mn−qλp , BdF (respectively IntF ) will denote the

    topological boundary (respectively interior) ofF in Cp.

    2. Motion along a submanifold and tubes of nonspherical section

    Definition 2.1. A motion fromp alongP is a family of diffeomorphisms{φx :Op ⊂ expp(W ∩NpP ) →Ox ⊂ expx(W ∩NxP )}x∈P , where theOx are open sets in expx(W ∩NxP ), satisfying:

    (a) eachφx∗p is an isometry of Euclidean vector spaces,(b) eachφx carries the geodesics throughp into geodesics throughx, with φp = Id, the identity map,

    and(c) the mapφ :P × expp(W ∩NpP ) → W defined byφ(x, ξ) = φx(ξ) is C∞.

    Taking ϕx = φx∗p, a motion can also be equivalently defined as a family of maps{ϕx :NpP →NxP }x∈P satisfying:

    (a′) eachϕx is a linear isometry,ϕp = Id, and(b′) the mapϕ :P × (NpP ) → NP defined byϕ(x, ξ) = ϕx(ξ) is C∞.

    The mapsφx andϕx are related by

    (2.1)ϕx = φx∗p and φx = expx ◦ϕx ◦ exp−1p .Obviously, the mapϕ is a vector bundle isomorphism, then the existence of a motion alosubmanifoldP implies that the normal bundle is trivial. This is not a breakdown for us, becausare interested in the computation of volumes, and, ifNP is not trivial, we can always subtract fromP asetZ of zero measure in order to haveP − Z be a submanifold with trivial normal bundle (neverthelesee the definition of tube of nonspherical section at the end of this section, and the remarks done

    Usually, the motion will be described either by the mapφ, or by the mapϕ of Definition 2.1.Let us remark that forM = Mnλ , the subsets expx(W ∩NxP ) are totally geodesic submanifolds ofMnλ ,

    and the mapsφx are isometries.Now, we shall consider a domainDp or a hypersurfaceCp contained in an open setOp of

    expp(W ∩ NpP ). Given a motionφ from p along P , we shall denote byD or C the domain or the

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 233

    rl,

    s

    al

    f

    lese

    hypersurface, respectively, obtained by the action of this motion onDp or Cp, that is

    D =⋃x∈P

    φx(Dp) = φ(P × Dp), C =⋃x∈P

    φx(Cp) = φ(P × Cp).

    The following example of motion will play an special role inSection 6. If every x ∈ P can be joinedto p by a unique minimizing geodesicα(t) of P with α(0) = p andα(�p(x)) = x, then we define themotion {ϕpx }x∈P by

    ϕpx (u) = Pp�p(x)(u) for everyu ∈NpP.This motion will be called theradial motion fromp.

    Given any motionϕ, every unit vectoru ∈ NpP defines a unit normal vector fieldNu on P byNu(x) = ϕx(u).

    A parallel motionis a motion satisfying that for everyx ∈ P , everyX ∈ TxP and every unit vectou ∈NpP , DX(Nu) = 0. This motion exists if and only if theD-holonomy of the normal bundle is triviaand, if so, it coincides with the radial motion.

    A tube (respectively a tubular hypersurface) of radiusr � c(P ) aroundP in M can be described athe image by exp of certain fiber subbundle ofNP by

    Pr = exp(NP(r)

    ) (respectively∂Pr = exp

    (SNP(r)

    )),

    where NP(r) = {v ∈ NP ; |v| � r} and SNP(r) = {v ∈ NP ; |v| = r}. That is, the tubePr(respectively the tubular hypersurface∂Pr ) is the image by the exponential map of a subbundle ofNPwith standard fiber a ball (respectively a sphere) ofRn−q . Then, if we look for tubes of nonsphericsection, the following is the natural

    Definition 2.2. Let D0 (respectivelyC0) be a domain (respectively a hypersurface) inRn−q . A tubePD(respectively a tubular hypersurfacePC ) of sectionD0 (respectivelyC0) around a submanifoldP of aRiemannian manifoldM is the image by the exponential map ofM of the subbundleD (respectivelyC)of NP with standard fiberD0 (respectivelyC0).

    It follows from this definition thatPD (respectivelyPC) exists if and only if there is a reduction othe structural groupO(n − q) of NP → P to the groupG = {ψ ∈ O(n − q) such thatψ(D0) = D0}(respectivelyψ(C0) = C0). In particular, ifD0 (respectivelyC0) has no symmetry,PD (respectivelyPC )only exists on submanifoldsP with trivial normal bundle.

    Let {(U,β)} be an atlas of the vector bundleπ :NP → P with transition functions inG. Then, everychartβ :π−1(U) → U × Rn−q defines, for eachp ∈ U , a map

    pϕ :U ×NpP → NP |U by pϕx(ξp) = β−1(x,π2

    (β(ξp)

    )),

    which is a motion alongU from p in the sense ofDefinition 2.1.Then, a tube of sectionD0 is locally obtained by the motion of someDp image by the exponentia

    of M of a Dp ⊂ NpP isometric toD0. A similar fact is true for tubular hypersurfaces. Thanks to thfacts, the formulae for volume(D) and volume(C) in Sections 3, 4, and 5are also valid for volume(PD)and volume(PC). There is also an obvious corresponding statement for tubes ofTheorem 6.3using thecharts ofNP .

    From now onM = Mnλ . Then theφx are isometries.

  • 234 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    r,s

    f

    nstance,

    3. A general formula to compute the volume of D in Mnλ

    The volume ofD can be computed following the arguments in[6] for the volume of a tube. Howevein order to have an unified approach to the computations of the volumes ofD andC (where the methodof [6] fail), we shall do a detailed derivation of a formula for volume(D).

    Theorem 3.1. Let D be the domain inMnλ obtained by the motionφ of a domainDp of Mn−qλp alongP .

    If dx anddz are, respectively, the volume elements ofP andDp, then

    volume(D) =∫P

    (∫Dp

    det(cλ

    (rp(z)

    )Id − sλ

    (rp(z)

    )LNφx(z)(x)

    )dz

    )dx.

    Proof. The motionφ restricted toP × Dp is a diffeomorphismφ :P × Dp → D which (cf. (2.1)) hasthe expression

    (3.1)φ(x,expp µ) = φx(expp µ) = expx ϕx(µ).Let {e1, . . . , eq} be an orthonormal basis ofTxP , and let {eq+1, . . . , en} be an orthonormal basis oTexpp µDp = τ|µ|NpP . Let us denote byω the volume element ofD (that is, the volume element ofM),then

    φ∗ω = φ∗ω(e1, . . . , eq, eq+1, . . . , en)dx ∧ dz= ω(φ∗e1, . . . , φ∗eq, φ∗eq+1, . . . , φ∗en)dx ∧ dz= |φ∗e1 ∧ · · · ∧ φ∗eq ∧ φ∗eq+1 ∧ · · · ∧ φ∗en|dx ∧ dz,

    where the last expression is the norm of the exterior product defined in the standard way (see, for i[10]).

    For 1� a � q, if ca is a curve inP satisfyingc′a(0) = ea,

    φ∗(x,expp µ)ea =d

    dtexpca(t) ϕca(t)(µ)

    is the value at|µ| of the Jacobi fieldYa(t) along expx tϕx(u), with u = µ/|µ|, satisfyingYa(0) = ea andY ′a(0) = ∇eaNu (cf. [12]), that is,φ∗ea = Ya(|µ|). It is easy to check that

    Ya(t) = cλ(t)τt ea + sλ(r)τt∇eaNu.On the other hand, sinceφx is an isometry and{eq+1, . . . , en} is an orthonormal basis ofTexpp µDp, thevectors

    ei := φ∗(x,expp µ)ei = φx∗expp µei, q + 1� i � nare an orthonormal basis ofTφx(expp µ)Dx .

    Then, from the skewsymmetry of the exterior product,

    (φ∗ω)(x,expp µ) =∣∣Y1(|µ|) ∧ · · · ∧ Yq(|µ|) ∧ eq+1 ∧ · · · ∧ en∣∣dx ∧ dz

    = ∣∣(cλ(|µ|)τ|µ|e1 + sλ(|µ|)τ|µ|(∇e1Nu)�) ∧ · · ·∧ (cλ(|µ|)τ|µ|eq + sλ(|µ|)τ|µ|(∇eqNu)�) ∧ eq+1 ∧ · · · ∧ en∣∣dx ∧ dz,

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 235

    d

    e

    e

    where (∇eaNu)� denotes the component of∇eaNu tangent toP . Having account that(∇eaNu)� =−LNuea and, again, the skewsymmetry of the exterior product, we have∣∣(cλ(|µ|)τ|µ|e1 + sλ(|µ|)τ|µ|(∇e1Nu)�) ∧ · · ·

    ∧ (cλ(|µ|)τ|µ|eq + sλ(|µ|)τ|µ|(∇eqNu)�) ∧ eq+1 ∧ · · · ∧ en∣∣dx ∧ dz= det(cλ(|µ|)Id − sλ(|µ|)LNu)|τ|µ|e1 ∧ · · · ∧ τ|µ|eq ∧ eq+1 ∧ · · · ∧ en|= det(cλ(|µ|)Id − sλ(|µ|)LNu).

    Since|µ| = rx(expx ϕx(µ)) = rp(expp µ), we have

    volume(D) =∫D

    ω =∫

    P×Dpφ∗(ω) =

    ∫P×Dp

    det(cλ

    (rp(z)

    )Id − sλ

    (rp(z)

    )LNφx(z)

    )dx ∧ dz

    which finishes the proof of the theorem.�

    4. Consequences of Theorem 3.1 and a Weyl type formula for motions

    An interesting consequence ofTheorem 3.1is that volume(D) depends only on the seconfundamental form ofP and not on the(k + 1)th fundamental forms fork � 2 (we call (k + 1)thfundamental form to thek + 1 symmetric covariant tensorsk defined in[13, p. 235 ff]). The nontrivialityof this statement will be clear when we show, inSection 5, that volume(C) generally depends on all th(k + 1)th fundamental forms.

    Developing the determinant in the formula for the volume(D) in Theorem 3.1, we have

    Corollary 4.1. Under the conditions ofTheorem3.1, if dy denotes the volume element ofDx ,

    volume(D) = volume(P )∫Dp

    cq

    λ

    (rp(z)

    )dz

    − q∫P

    Hx

    (∫Dx

    cq−1λ

    (rx(y)

    )sλ

    (rx(y)

    )〈h,Ny(x)

    〉dy

    )dx

    +q∑

    i=2(−1)i

    (q

    i

    )∫P

    (∫Dx

    siλ(rx(y)

    )cq−iλ

    (rx(y)

    )HiNy(x) dy

    )dx,

    whereHiNy(x) is the ith mean curvature ofP in the direction ofNy(x), andHx (respectivelyh) is thenorm (respectively the unit vector in the direction) of the mean curvature vector ofP at x.

    As a function ofu = Ny(x), HiNy(x) is a homogeneous polynomial of degreeith onu, then,if Dp iscentral symmetric with respect top, the terms corresponding toi odd vanish, and volume(D) dependsonly on theith mean curvatures ofP of order even. Now we are going to see that, for generalDp, we canstill choosep in Dp such that volume(D) does not depend onH := H 1. For it, we generalize slightly thnotion of centre of mass by that ofq-centre of mass.

  • 236 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    r

    e foreometryyrue,gs

    on

    ce

    Forλ = 0, we still follow the classical definition: “theq-centre of mass ofB is the pointx0 ∈ Mn−qλ onwhich the functionp → ∫B d(p, z)2 dz attains its minimum value”. Whenλ �= 0, we define theq-centreof massas the pointx0 where the function

    F :p → −λ∫B

    cq

    λ

    (rp(z)

    )dz

    attains its minimum value (whenq = 1, this is the center of mass defined in[8]). For everyζ ∈ TpMn−qλp ,

    (4.2)〈gradF(p), ζ

    〉 = −qλ2 ∫B

    cq−1λ

    (rp(z)

    )sλ

    (rp(z)

    )〈Nz(p), ζ

    〉dz,

    which has to vanish ifp is a minimum. Moreover

    HessF(p)(X,X) = qλ2∫B

    cq−2λ

    (rp(x)

    )(c2λ

    (rp(x)

    ) − λ(q − 1)s2λ(rp(x)))〈X,∂r〉2 dx(4.3)+ qλ2

    ∫B

    cq

    λ

    (rp(x)

    )|X⊥|2 dx,where∂r is the gradient of the functionrp and X⊥ is the component ofX orthogonal to∂r . Then, astandard argument (see, for instance,[1, Chapter 8]or [9]) shows that theq-centre of mass exists ifBhas compact closure (and is contained in a ball of radius12z+(c

    2λ(r(x)) − λ(q − 1)s2λ(r(x)) if λ > 0).

    It follows from this definition and formula ofCorollary 4.1for volume(D) that if p is theq-centre ofmass ofDp, thenvolume(D) does not depend on the mean curvature ofP , it depends only on the higheorder mean curvatures. This generalizes Theorem 1 and its corollary in[7] to higher dimensions.

    The situation looks different from that of Weyl’s Theorem (and from that of Pappus formulamotions along curves), where the volume of a tube depends only on the radius and the intrinsic gof P . Here, the analogous result would be that volume(D) depends only onDp and the intrinsic geometrof P , but, using formula ofCorollary 4.1, it is easy to find examples showing the this is no longer teven if we takeDp central symmetric. For instance, ifI = ]0,2π [, we consider the isometric embeddinji : I 2 → R4, i = 1,2, defined byj1(u, v) = (u, v,0,0) andj2(u, v) = (cosu,sinu,cosv,sinv), Dp =]−a,0[2 ∪ ]0, a[2, we putDp on the normal bundle of eachji(I 2) with the point(0,0) at the 0 sectionand with the canonical basis ofR2 identified with the basis{(0,0,1,0), (0,0,0,1)} for i = 1, and with{(cosu,sinu,0,0), (0,0,cosv,sinv)} for i = 2. If Di is the domain obtained by the parallel motiof Dp alongji(I 2), then volume(D1) �= volume(D2), althoughj1(I 2) andj2(I 2) are isometric (but withdifferent second fundamental form).

    However, we still have an analogue of Weyl’s Theorem ifDp has enough symmetries. Let us introduthe notation

    I (i1, j1, . . . , is, js) =∫Dp

    sbλ(rp(z)

    )cq−bλ

    (rp(z)

    )Nz(p)

    j1i1

    · · ·Nz(p)jsis dz,

    0� j1 + · · · + js = b � q,whereNz(p)i denotes theith coordinate of the unit vectorNz(p) in a given orthonormal basis ofNpP .

    From now on, a chain of inequalities of the formi1 �= · · · �= is will mean thatik �= i� if k �= �.

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 237

    sicnal

    n

    Definition 4.2. We say that a domainDp in Mn−qλp is q-symmetric with respect top if there is an

    orthonormal basis{eq+1, . . . , en} of NpP such that following identities are satisfied:

    (i) I (i1, j1, . . . , is, js) = 0 if i1 �= · · · �= is and some of thej1, . . . , js are odd, and(ii) I (i1,2j1, . . . , is,2js) = 2j1)...2js)2b) I (1,2b) if i1 �= · · · �= is and 1� j1 + · · · + js = b � q/2, where

    2j) = 1 3 5. . . (2j − 1).

    Let us remark thatDp is 1-symmetric with respect top if and only if p is the 1-centre of mass ofDp.The next proposition gives a big family of examples ofq-symmetric domains which are not geode

    spheres. Given a basis{eq+1, . . . , en} of NpP , let Πi be the hyperplane through the origin orthogoto ei , and letΠik be the subspace through the origin of dimensionn − q − 2 and orthogonal toeiand ek . By a rotation of axisΠik we shall understand an isometry ofNpP preserving the orientatioand havingΠik as the set of fixed points.

    Proposition 4.3. If a domainDp in Mn−qλp satisfies that there is an orthonormal basis{eq+1, . . . , en} of

    NpP such that

    (i) exp−1p (Dp) is invariant by each one of then − q symmetries with respect toΠi, q + 1� i � n, and(ii) exp−1p (Dp) is invariant by each one of the rotations with axisΠik and angleπ/(2b), b = 1, . . . , [q/2],

    with q + 1 � i, k � n, i �= k,

    thenDp is q-symmetric with respect top.

    Proof. The condition (i) inDefinition 4.2follows from the condition (i) of this proposition.Let Rijθ be the rotation induced onM

    n−qλp by the rotation of axisΠik and angleθ in NpP . Since

    det(Rikθ∗) = 1 andrp(Rikθ (z)) = rp(z),∫Rikθ (Dp)

    s2bλ(rp(z

    ′))cq−2bλ

    (rp(z

    ′))Nz′(p)

    2j1i1

    · · ·Nz′(p)2jsis dz′

    =∫Dp

    s2bλ(rp(z)

    )cq−2bλ

    (rp(z)

    )NRikθ (z)(p)

    2j1i1

    · · ·NRikθ (z)(p)2jsis dz.

    Let α = cosθ , β = sinθ , then, for 1� � � s,

    NRikθ (z)(p)i� =

    Nz(p)i� if i� /∈ {i, k},αNz(p)i − βNz(p)k if i� = i,βNz(p)i + αNz(p)k if i� = k,

    then if Dp is invariant byRikθ , from the above integral equality and condition (i) ofDefinition 4.2, wehave, for everyi, k, with k /∈ {i1, . . . , i, . . . , is},

    I (i1,2j1, . . . , i,2j�, . . . , is,2js)

    =∫D

    s2bλ(rp(z)

    )cq−2bλ

    (rp(z)

    )Nz(p)

    2j1i1

    · · · (αNz(p)i − βNz(p)k)2j� · · ·Nz(p)2jsis dz

    p

  • 238 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    t

    t

    (4.4)=j�∑

    r=0

    (2j�2r

    )α2j�−2rβ2r I (i1,2j1, . . . , i,2j� − 2r, . . . , k,2r, . . . , is,2js).

    SinceDp satisfies the condition (ii) of this proposition,(4.4) holds for θc = π/(2c), αc = cosθc andβc = sinθc, c = 1, . . . , j�. Then, using that(α2c + β2c )j� = 1, we get the equations

    j�∑r=1

    α2j�−2rc β2rc

    ((2j�2r

    )I (i,2j� − 2r, k,2r) −

    (j�r

    )I(i,2j�)

    )= 0, c = 1, . . . , j�,

    whereI (i,2j� − 2r, k,2r) := I (i1,2j1, . . . , i,2j� − 2r, . . . , k,2r, . . . , is,2js) and I(i,2j�) = I (i1,2j1,. . . , i,2j�, . . . , is,2js). This is a homogeneous system ofj� linear equations in thej� unknowns(2j�

    2r

    )I(i,2j� − 2r, k,2r) −

    (j�r

    )I (i,2j�), r = 1, . . . , j�. Using the notationac = (βc/αc)2, the determinan

    of the coefficients is∣∣∣∣∣∣∣∣∣0 . . . 0 1

    α2j�−22 β

    22 . . . α

    22β

    2j�−22 β

    2j�2

    .... . .

    ......

    α2j�−2j�

    β2j� . . . α2j�β

    2j�−2j�

    β2j�j�

    ∣∣∣∣∣∣∣∣∣ = (−1)j�+1α2j�2 · · ·α2j�j� a2 · · ·aj�

    j�−2∏k=1

    j�−k∏i=2

    (ai+k − ai) �= 0,

    then (2j�2r

    )I(i,2j� − 2r, k,2r) −

    (j�r

    )I(i,2j�) = 0,

    and

    I(i,2j� − 2r, k,2r) = 2r)2(j� − r))2j�) I(i,2j�).By recurrence,

    I (i1,2j1, . . . , is−1,2js−1, is,2js) = 2js−1)2js)2(js−1 + js))I

    (i1,2j1, . . . , is−2,2js−2, is−1,2(js−1 + js)

    )= · · · = 2js)2js−1)2js−2) · · ·2j1)

    2(j1 + · · · + js)) I(i1,2(j1 + · · · + js)

    ).

    On the other hand, using the invariance ofDp by a rotation of angleπ2 , it follows the equality

    I (i1,2b) = I (1,2b). Then, condition (ii) ofDefinition 4.2is also satisfied. �In the next theorem, we shall use the following notation, taken from[5].If RP andRM

    nλ are the curvature tensors ofP andMnλ , respectively, andAi , Bi are vectors tangen

    to P ,(RP − RMnλ )s(A1 ∧ · · · ∧ A2s)(B1 ∧ · · · ∧ B2s)=

    ∑π,σ∈Q̃s

    �π�σ(RP − RMnλ )

    Aπ(1)Aπ(2)Bσ(1)Bσ(2)· · · (RP − RMnλ )

    Aπ(2s−1)Aπ(2s)Bσ(2s−1)Bσ(2s) ,

    whereQ̃s = {σ ∈ S2s; σ (2t − 1) < σ(2t), t = 1, . . . , s}.

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 239

    r the

    The contractionCt , 0� t � 2s, is defined inductively byC0(Rs) = Rs andCt

    ((RP − RMnλ )s)(A1 ∧ · · · ∧ A2s−t )(B1 ∧ · · · ∧ B2s−t )

    =q∑

    a=1Ct−1

    ((RP − RMnλ )s)(A1 ∧ · · · ∧ A2s−t ∧ Ea)(B1 ∧ · · · ∧ B2s−t ∧ Ea),

    whereEa is a local orthonormal frame ofP .Finally, the invariantk2b(RP − RMnλ ) is defined by

    k2b(RP − RMnλ ) = 1

    b!(2b)!∫P

    C2b((

    RP − RMnλ )b)dz.Whenλ = 0, thek2b are, up to some constants, the coefficients in the classical Weyl’s formula fovolume of a tube, and it is a multiple of Euler characteristic ofP when 2b = q.

    Theorem 4.4. Under the conditions ofTheorem3.1, if Dp is q-symmetric with respect top, then

    (4.5)volume(D) =[q/2]∑b=0

    d2b(Dp)k2b(RP − RMnλ ),

    where

    (4.6)d2b(Dp) = 2b

    (2b)!∫Dp

    s2bλ(rp(z)

    )cq−2bλ

    (rp(z)

    )Nz(p)

    2b1 dz.

    Proof. We can write the formula inCorollary 4.1under the form

    volume(D) =q∑

    s=0

    ∫P

    (∫Dx

    ssλ(rx(y)

    )cq−sλ

    (rx(y)

    )Ψsx

    (Ny(x), . . . ,Ny(x)

    )dy

    )dx,

    whereΨs is the symmetrics-tensor field satisfying

    (−1)s(

    q

    s

    )HsNy(x) = Ψsx

    (Ny(x), . . . ,Ny(x)

    ).

    Let Ψ i1...issx be the components ofΨsx in an orthonormal basis{ϕx(eq+1), . . . , ϕx(en)}.At anyy = φx(z), we haveNy(x) = Nφx(z)(φx(p)) = ϕx(Nz(p)),

    Ny(x) =n∑

    i=q+1Ny(x)iϕx(ei) = ϕx

    (Nz(p)

    ) = ϕx(

    n∑i=q+1

    Nz(p)iei

    ),

    thenNy(x)i = Nz(p)i , and∫ssλ

    (rx(y)

    )cq−sλ

    (rx(y)

    )Ψsx

    (Ny(x), . . . ,Ny(x)

    )dy

    Dx

  • 240 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    =n∑

    i1,...,is=q+1

    ∫φx(Dp)

    ssλ(rx(y)

    )cq−sλ

    (rx(y)

    )Ψ i1...issx Ny(x)i1 · · ·Ny(x)is dy

    =n∑

    i1,...,is=q+1

    ∫Dp

    ssλ(rp(z)

    )cq−sλ

    (rp(z)

    )Ψ i1...issx Nz(p)i1 · · ·Nz(p)is dz,

    therefore

    volume(D) =q∑

    s=0

    ∫P

    ∑i1,...,is

    Ψ i1...issx

    (∫Dp

    ssλ(rp(z)

    )cq−sλ

    (rp(z)

    )Nz(p)i1 · · ·Nz(p)is dz

    )dx.

    SinceDp is q-symmetric,I (i1, j1, . . . , is, js) = 0 whenj1 + · · · + js is odd. Then

    volume(D) =[q/2]∑b=0

    ∫P

    ∑i1,...,i2b

    Ψi1...i2b2bx I (i1,1, . . . , i2b,1)dx,

    but, using again thatDp is q-symmetric,∑i1,...,i2b

    Ψi1...i2b2bx I (i1,1, . . . , i2b,1)

    =∑

    i1 �=···�=isj1+···+js=b

    (2b)!(2j1)! · · · (2js)!k1! · · · kr !Ψ

    i1

    2˘j1... i1...is

    2˘js

    ... is2bx I (i1,2j1, . . . , is,2js),

    wherek1, . . . , kr are the numbers such thatj�1 = · · · = j�k1 , . . . , jm1 = · · · = jmkr .But, from theq-symmetry ofDp, using (ii) ofDefinition 4.2, we have∑

    i1,...,i2b

    Ψi1...i2b2bx I (i1,1, . . . , i2b,1)

    = I (1,2b)∑

    i1 �=···�=isj1+···+js=b

    (2b)!(2j1)! · · · (2js)!k1! · · ·kr !

    2j1) · · ·2js)2b)

    Ψi1

    2˘j1... i1...is

    2˘js

    ... is2bx

    = I (1,2b)∑

    i1 �=···�=isj1+···+js=b

    (b)!j1! · · · js!k1! · · · kr !Ψ

    i1

    2˘j1... i1...is

    2˘js

    ... is2bx = I (1,2b)

    ∑i1,...,ib

    Ψi1i1...ibib2bx ,

    but it is proved in[5, pp. 223–224, before formula (7.10)]that

    ∑i1,...,ib

    Ψi1i1...ibib2bx =

    2b

    ((2b)!)2C2b

    (RP − RM)b

    x,

    and the formula in the theorem follows from the combination of these formulae.�

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 241

    at

    torits

    5. A formula for volume(C) in Mnλ

    Theorem 5.1. Let C be a hypersurface inMnλ obtained by the motionφ of a hypersurfaceCp in Mn−qλp

    along a submanifoldP of dimensionq. Letdx anddz be, respectively, the volume elements ofP andCp,and let{ea}qa=1 be an orthonormal basis ofTxP diagonalizing the Weingarten mapLNφx(z)(x). Then,

    (5.1)volume(C) =∫P

    ( ∫Cp

    √√√√(det(cλI − sλLN))2 + s2λ q∑a=1

    D2a dz

    )dx,

    where

    Da = det(cλI − sλLN)|e⊥a 〈τtDeaN, ξ̄〉,|e⊥a denotes the restriction to the subspace ofTxP orthogonal toea , and we take the convention thdet(cλI − sλLN)|e⊥a = 1 if q = 1.Proof. Using the motionφ restricted toP × Cp, if η is the volume element ofC, we have

    volume(C) =∫C

    η =∫

    P×Cpφ∗η.

    If {e1, . . . , eq} is an orthonormal basis ofTxP satisfying dx(e1, . . . , eq) = 1, and{eq+2, . . . , en} is anorthonormal basis ofTzCp satisfying dz(eq+2, . . . , en) = 1, then, arguing like in the proof ofTheorem 3.1and using the same notation, with the only exception that now∧ denotes the cross vector product inMnλ ,

    (5.2)(φ∗η

    )(x,expp µ)

    = ∣∣Y1(|µ|) ∧ · · · ∧ Yq(|µ|) ∧ eq+2 ∧ · · · ∧ en∣∣dx ∧ dz,whereei = φx∗z(ei). Then, we only have to computeY1 ∧ · · · ∧ Yq ∧ eq+2 ∧ · · · ∧ en.

    If we also use∧ to denote the cross vector product in exp(NxP ), andξ is the unit vector normal toCpat z, then

    (5.3)ξ = eq+2 ∧ · · · ∧ en, and ξ̄ := φx∗z(ξ) = eq+2 ∧ · · · ∧ enis the unit vector normal toCx in φx(z).

    It is easy to check that the Jacobi fieldsYa have the form

    (5.4)Ya(t) = Xa(t) + Da(t),with Xa(t) = cλ(t)τt ea − sλ(t)τtLuea ∈ τtTxP andDa(t) = sλ(t)τtDea (Nu) ∈ τtNxP .

    Then

    Y1 ∧ · · · ∧ Yq ∧ eq+2 ∧ · · · ∧ en = (X1 + D1) ∧ · · · ∧ (Xq + Dq)eq+2 ∧ · · · ∧ en= X1 ∧ · · · ∧ Xq ∧ eq+2 ∧ · · · ∧ en + X1 ∧ · · · ∧ Xq−1 ∧ Dq ∧ eq+2 ∧ · · · ∧ en + · · ·

    (5.5)+ D1 ∧ X2 ∧ · · · ∧ Xq ∧ eq+2 ∧ · · · ∧ en + (terms with twoDa or more).From (5.3), it follows that eq+2, . . . , en, ξ̄ is an orthonormal basis ofτtNxP , and theDa are in this

    space, then the terms in(5.5) with two or moreDa vanish (from the skewsymmetry of the cross vecproduct), and, in the terms with only oneDa, Da contributes to a nonzero summand only with

  • 242 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    n bering,

    component〈Da, ξ̄〉ξ̄ . ThenY1 ∧ · · · ∧ Yq ∧ eq+2 ∧ · · · ∧ en

    = X1 ∧ · · · ∧ Xq ∧ eq+2 ∧ · · · ∧ en(5.6)+

    q∑a=1

    〈Da, ξ̄〉X1 ∧ · · · ∧ ξ̄(a) ∧ · · · ∧ Xq ∧ eq+2 ∧ · · · ∧ en.

    Takinge1, . . . , eq eigenvectors ofLN , with corresponding eigenvaluesk1, . . . , kq , and using(5.4)and theproperties of the cross vector product, we obtain

    X1 ∧ · · · ∧ Xq ∧ eq+2 ∧ · · · ∧ en= (cλ − k1sλ)e1 ∧ · · · ∧ (cλ − kqsλ)eq ∧ eq+2 ∧ · · · ∧ en

    (5.7)= (cλ − k1sλ) · · · (cλ − kqsλ)ξ̄ = det(cλI − sλLN)ξ̄and

    〈Da, ξ̄ 〉X1 ∧ · · · ∧ ξ̄(a) ∧ · · · ∧ Xq ∧ eq+2 ∧ · · · ∧ en

    = 〈Da, ξ̄〉{(cλ − k1sλ)e1 ∧ · · · ∧ ξ̄

    (a)∧ · · · ∧ (cλ − kqsλ)eq ∧ eq+2 ∧ · · · ∧ en

    }= sλ

    〈τtDea (u), ξ̄

    〉{(cλ − k1sλ)e1 ∧ · · · ∧ ξ̄(a) ∧ · · · ∧ (cλ − kqsλ)eq ∧ eq+2 ∧ · · · ∧ en

    }= (−1)n−asλ

    〈τtDea (u), ξ̄

    〉(cλ − k1sλ) · · · (cλ − ka−1sλ)(cλ − ka+1sλ) · · · (cλ − kqsλ)ea

    (5.8)= (−1)n−asλ〈τtDea (u), ξ̄

    〉det(cλI − sλLu)|e⊥a ea.

    From(5.2), (5.6), (5.7), (5.8)and the expression for volume(C), the theorem follows. �Remark. Following the notation in[13, Vol. 4, Chapter 7], if P is a nicely embedded submanifold ofMnλ ,any unit normal vectorN can be decomposed as

    N = N1 + · · · + N�, with Nk ∈ Nork P andDeaNk = −AkNk (ea) + DkeaNk + sk(ea,Nk),

    with the convention thatA1N1(ea) = 0 ands�(ea,N�) = 0.Then, in general, all the(k + 1)th fundamental forms ofP appear in the formula for volume(C),

    a situation very different from that of volume(D) that we considered before. This dependence cachecked taking helixh1 andh2 in R3 with the same curvature and different torsion, and considein R4, the surfacesh1 × R andh2 × R.

    6. When does volume(C) attain its lower bound?

    From the formula ofTheorem 5.1we have

    (6.1)volume(C) �∫

    P×Cdet

    (cλ

    (rp(z)

    )I − sλ

    (rp(z)

    )LNφx(z)(x)

    )dx ∧ dz

    p

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 243

    dge ofn

    e

    ic

    fold

    desicchen

    and we have the equality for a parallel motion. In this section we are interested on the knowleconditions, other than parallel motion, giving the equality in(6.1). Let us also remark that such aequality implies that volume(C) depends only on the second fundamental form ofP and not on thesuccessive ones.

    For a hypersurfaceCp of Mn−qλp it is possible to define the notion ofq-symmetry with respect top, just

    changingDp by Cp everywhere inDefinition 4.2. In general, the lower bound given by(6.1)depends onthe motionφ and the second fundamental form ofP , but,whenCp is q-symmetric with respect top, thesame computations done in the proof ofTheorem 4.4show thatthere is the corresponding Weyl’s typformula for the right hand of inequality(6.1) (just change everywhereDp by Cp in formula(4.6)and inthe right hand of formula(4.5)), then it does not depend on the motion,and it is an intrinsic invariantof P andCp.

    We shall see that, for most motions, the equality in(6.1)only holds whenCp is contained in a geodessphere ofMn−qλp , and, for many others, the equality in(6.1) implies thatCp is a revolution submanifold.

    First, we shall give technical conditions on the motion and the hypersurfaceCp characterizing theequality in(6.1). Then we shall describe more geometrically these conditions.

    A hypersurfaceH of Mn−qλp is called arevolution hypersurface around a totally geodesic submaniV of dimensiond < n − q − 1 if H is invariant by the natural action ofSO(n − q − d) on Mn−qλp whichhas the submanifoldV as the set of fixed points. This is equivalent to say that, for every totally geosubmanifoldM of dimensionn−q −d orthogonal toV , the intersectionM∩H is the union of geodesispheres ofM with centre inM ∩ V (including the geodesic spheres of radius 0). In particular, wd = 0,V is a pointp and a revolution hypersurface aroundp is just a geodesic sphere with centre atp.

    From now on, given a vector subspaceV of NpP , andz ∈ expp V = V , by Mn−q−dλz we shall denotethe totally geodesic submanifold ofMn−qλp of dimensionn−q −d throughz and orthogonal toV if λ � 0,and the open ball of radiusπ

    2√

    λand centrez of this submanifold whenλ > 0.

    Lemma 6.1. Letϕ be a motion alongP from p satisfying that the equality holds in(6.1) and

    there is a vector subspaceV ofNpP of dimensiond � n − q − 2such that for everyy ∈ Cp − expp V (that is, for everyu = Ny(p) ∈ NpP − V )there aren − q − d − 1 pointsxq+d+2, . . . , xn ∈ P and vectorsXi ∈ TxiPsuch that the vectors(ϕxi )

    −1DXiNu(xi), i = q + d + 2, . . . , n,(6.2)are linearly independent and orthogonal toV,

    then

    (a) whend = 0, Cp is contained in a geodesic sphere inMn−qλp . If, moreover,Cp is closed(compactwithout boundary), then it is a sphere;

    (b) whend > 0 and Cp is closed(and, whenλ > 0, dist(z,expp V ) <π

    2√

    λfor everyz ∈ Cp), Cp is a

    revolution hypersurface aroundexpp V .

    Proof. The equality in(6.1) holds if and only if(det(cλI − sλLNu)|e⊥a )〈τtDea (Nu), ξ̄〉 = 0 for everya,everyu and everyr satisfying expx rϕx(u) ∈ Cx , and, since we are before the focal points ofP , this isequivalent to

    (6.3)〈τtDea (Nu), ξ̄

    〉 = 0 for everyu and everya,

  • 244 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    t

    ormale

    l

    and this implies

    (6.4)〈τtDXi (Nu), ξ̄

    〉 = 0 for everyu and everyi.Moreover, sincex �→ Nu(x) = ϕx(u) is a unit vector field,

    (6.5)〈DXiNu,Nu〉 = 0,and, sinceϕx is an isometry, we get from(6.4)and(6.5) that〈

    ϕ−1xi (DXiNu)xi , ξp〉 = 0 and

    (6.6)〈ϕ−1xi (DXNu)xi , u

    〉 = 0 for i = q + d + 2, . . . , n,whereξx = τ−1t ξ̄x . If d = 0, (6.2)and(6.6) imply thatξp = u, that is,ξ̄p = gradrp, and this implies thaCp is contained in a geodesic sphere. This finishes the part (a) of the lemma.

    In order to prove the part (b), we shall show that the nonempty intersections ofCp with Mn−q−dλz ,

    z ∈ expp V , are union of geodesic spheres ofMn−q−dλz .From the hypotheses,Cp is contained in an open set where the exponential map on the n

    bundleNV to V in Mn−qλp is a diffeomorphism. Then, every point inCp can be written, in a uniquway, as expz w, with z ∈ V andw ∈ NzV . The mapf :Cp → V defined byf (expz w) = z is C∞, andCp ∩ Mn−q−dλz = f −1(z). Of course, the mapf has a well definedC∞ extensionf̄ on allMn−qλp (or Mn−qλpwithout the cut points ofV if λ > 0), defined by the same expression.

    For everyz = expz w ∈ f −1(z), let γ (s) = expz s(w/|w|) be the geodesic fromz to γ (|w|) = z. IfYa(s), a = q +1, . . . , q +d are the Jacobi fields alongγ (s) satisfying that{Ya(0)}q+da=q+1 is an orthonormabasis ofT zV andY ′a(0) = 0; andJi(s), i = q + d + 1, . . . , n, are the Jacobi fields alongγ (s) satisfyingthatJi(0) = 0 and{J ′i (0)}ni=q+d+1 is an orthonormal basis ofNzV , then

    (6.7)f̄∗zYa(|w|) = Ya(0) and f̄∗zJi(|w|) = 0.

    Let Ps be the parallel transport along the geodesicγ (s). SinceMn−qλp has constant sectional curvature,

    (6.8)Ya(0) is in the direction of P−1s Ya

    (|w|).Sincef is the restriction off̄ to Cp, it follows from (6.7)and(6.8) that

    (6.9)z is a not regular point off if and only if ξ̄z ∈ PsTzV.Now, let us suppose thatz is a regular value off . Then, every connected componentCz of f −1(z) is a

    compact regular submanifold ofCp of dimensionn − q − d − 1, and, for everyz ∈ Cz ⊂ Cp ∩ Mn−q−dλz ,ξ̄z /∈ PsTzV , and, forz �= z, theu ∈NpP such thatz = expp rp(z)u is not inV .

    Let U be the totally geodesic submanifold of dimensiond + 1 containingV and tangent tou at p. Itis obvious that〈u〉 ⊕ V = TpU , where〈u〉 denotes the vector space generated byu.

    From(6.2)and(6.6), there aren−q −d −1 linearly independent vectorsvi orthogonal toξp, V andu.Then, sinceu /∈ V , from the dimensions it follows thatξz ∈ TpU , and, taking the parallel transportτt ,sinceU is a totally geodesic submanifold,ξ̄z ∈ τtTpU = TzU . Since the geodesicγ (s) is contained inU ,PsTzV ⊂ TzU , and sincēξz /∈ PsTzV , we have

    TzU = 〈ξ̄z〉 ⊕ PsTzV.

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 245

    esof

    ank

    tl

    n.

    ing

    t

    ints

    Given thatξ̄z is orthogonal toCp at z andPtTzV is orthogonal toMn−q−dλz at z, we have thatTzU =〈ξ̄z〉 ⊕ PtTzV is orthogonal toCz ⊂ Mn−q−dλz ∩ Cp. On the other hand, the gradient inMn−q−dλz of thedistance toz is tangent to the geodesicγ (s), which is contained inU , then it is orthogonal toCz. thereforeCz is a geodesic sphere ofMn−q−dλz with centre atz or Cz = {z}.

    Now, let us suppose thatz is a critical value off . From Sard’s Theorem, the set of critical valuof f has zero measure. LetZ be the arcwise connected component of the set of critical valuesfcontainingz. Let CZ be an arcwise connected component off −1(Z).

    If Bd CZ = ∅, thenCZ is open and closed inCp, thenCZ = Cp, and this impliesCp = f −1(Z). If therebe some pointy ∈ Cp regular forf , it would have a neighborhood of regular points, and, by the rconstant theorem, its image would be a neighborhood off (y), which is impossible becausef (Cp) = Zhas zero measure. ThenCp has no regular point off . Thus, from(6.9)we have that

    (6.10)ξ̄z ∈ PsTzV for everyz ∈ Z and everyz ∈ f −1(z).SinceCp is compact, there is a pointz ∈ Cp at maximal distance fromV , and, at this point, the gradienin Cp of the distanced in M

    n−qλp toV must vanish, then, the gradient ofd in M

    n−qλp atz must be orthogona

    to Cp, that is, in the direction of̄ξz, in contradiction with(6.10). Thenwe can conclude thatBdCZ �= ∅.Now, letz ∈ BdCZ. Every connected neighborhoodU of z in Cp has pointszi such thatf (zi) /∈ Z.

    In fact, if U ⊂ f −1(Z), sincez ∈ BdCZ, U ∩ (Cp − CZ) �= ∅, and, sinceU is connected, the points iU − CZ could be united toz by an arc inU , then inf −1(Z), thereforeU ⊂ CZ, which is a contradiction

    So then, givenz ∈ BdCZ, and the family{B(z,1/n)}n∈N of open balls inMn−q−dλf (z) centered atz withradius 1/n, we consider a family of connected neighborhoodsUn of z in Cp satisfyingUn ⊂ B(z,1/n).From the above remark, for everyn there is az′n ∈ Un such thatf (z′n) /∈ Z. Let cn be a curve inUnjoining z′n andz. The image ofcn by f will be a curve inV joining f (z′n) /∈ Z with f (z) ∈ Z. SinceZ isan arcwise connected component of the set of critical values, there must be a regular valuef (zn) in theimage byf of the curvecn. Take a pointzn in this curve which image isf (zn), thenzn ∈ Un ⊂ B(z,1/n)and the sequence{zn}∞n=1 converges toz. From the above discussion on regular values,zn is in somegeodesic sphereSn of M

    n−q−dλf (zn)

    with centre atf (zn) ∈ V and contained inCp.For every p1 ∈ S1, we construct a sequence of points defined inductively in the follow

    way: given pn ∈ Sn, we take pn+1 as the point inSn+1 closest topn. Then dist(pn,pn+1) =dist(Sn,Sn+1), and the sequence{pn} satisfies dist(pn,pn+1) � dist(zn, zn+1), then it is convergento some pointp0 ∈ Cp (becauseCp is closed). Moreover, for everyn, f (pn) = f (zn), and f (z) =limn→∞ f (zn) = limn→∞ f (pn) = f (p0) for every p1 ∈ S1, and we also have that dist(p0, f (z)) =limn→∞ dist(pn, f (zn)) = limn→∞ dist(zn, f (zn)), then, independent onp1. Then, all the limits of thesequences constructed above for everyp1 ∈ S1 are contained inf −1(f (z)) ⊂ Mn−q−dλf (z) and in thegeodesic sphereS of centref (z) and radius dist(f (z), z). On the other hand, all the points inS arelimit of one of these sequences: givenp0 ∈ S, we can construct the sequence whichnth element is thepoint pn ∈ Sn closest top0. Then, the points inS are limits of sequences with points inSn, soS ⊂ Cp.SinceCp is compact, andS ⊂ CZ, because it is inf −1(f (z)) ⊂ f −1(Z) and all points inS can be unitedto z by an arc. Moreover, all the points inS are in BdCZ, because they are limits of sequences of powith regular image byf . Then, all points in BdCZ are contained in geodesic spheres ofM

    n−q−dλz′ , with

    z′ ∈ Z, and each one of these geodesic spheres is contained in BdCZ. Then, BdCZ is the union of somegeodesic spheres ofMn−q−dλz′ , with z

    ′ ∈ Z.Now, lety be an interior point ofCZ in Cp. First two remarks:

  • 246 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    ed

    as

    s strictlyhe

    sphere

    Remarks.

    (i) Again by the rank constant theorem and the fact thatf (CZ) ⊂ Z has measure zero,y is not a regularpoint off .

    (ii) Bd CZ ∩ Mn−q−dλf (y) �= ∅.

    In fact, first we shall show that IntCZ ∩ Mn−q−dλf (y) is an open set inMn−q−dλf (y) . We know thatTy IntCZ =TyCp ⊂ TyMn−q−dλf (y) ⊕ PsTf (y)V . Sincey is not a regular point off , (TyCp)⊥ ⊂ PsTf (y)V , then, from thedimensions,TyM

    n−q−dλf (y) ⊂ TyCp = Ty IntCZ. Theny �→ TyMn−q−dλf (y) defines a(n − q − d)-dimensional

    distribution∆ on IntCZ which is the restriction to IntCZ of the distribution∆ defined onMn−qλp by the

    same expression, and any vector fieldX on IntCZ tangent to∆ can be extended to a vector fieldX definedon a tubular neighborhood of IntCZ and tangent to∆ by Xy ′ = Ps ◦ τf (y ′)f (y) ◦ P −1s Xy , wherey is the pointof IntCZ nearest toy′ andτ

    f (y ′)f (y) is the parallel transport along the minimizing geodesic joiningf (y) and

    f (y′). Then∆ is involutive, and its integral leaves are open sets ofMn−q−dλz′ for z′ ∈ f (CZ), then every

    y ∈ IntCZ ∩ Mn−q−dλf (y) is contained in an open set ofMn−q−dλf (y) which is contained in IntCZ ∩ Mn−q−dλf (y) , thenIntCZ ∩ Mn−q−dλf (y) is open inMn−q−dλf (y) . If Bd CZ ∩ Mn−q−dλf (y) = ∅, thenCZ ∩ Mn−q−dλf (y) = IntCZ ∩ Mn−q−dλf (y) isopen and closed inMn−q−dλf (y) , thenCZ ∩ Mn−q−dλf (y) = Mn−q−dλf (y) which is impossible becauseCp (thenCZ) iscompact.

    From remark (ii) and the above study on the points in BdCZ it follows that BdCZ ∩Mn−q−dλf (y) is a unionof geodesic spheres, then there are two geodesic spheresS,S′ ⊂ BdCz ∩ Mn−q−dλf (y) (perhaps withS′ = ∅or just one point) satisfying thaty ∈D = IntB−B′, whereB andB′ are the closed geodesic balls boundby S andS′ respectively, and there is no geodesic sphere ofMn−q−dλf (y) contained in BdCZ ∩ Mn−q−dλf (y)betweenS and S′. Let O be the maximal connected open set inCp containingy and contained inf −1(Z). If there is a pointy′ ∈ D − O, then there must be a point in BdCZ ∩ D because, if not, theexteriorE of O must containy′, andD can be decomposed as the union of the two open setsE ∩D andO ∩D, and it will not be connected. Then, all the points inD are inCZ. Then,CZ is the union of somegeodesic spheres of someMn−q−dλz . This finishes the proof of the lemma.�

    Remark. The first reason for the separation of casesd = 0 andd > 0 in the above and the next lemmand theorems is technical: we used the hypothesis of closedness ofCp only whend > 0. The interest forwriting a statement without the closedness hypothesis resides on the existence of hypersurfacecontained on a geodesic sphere which areq-symmetric, which gives tubes with holes for which tWeyl’s type formula is still valid. An easy example of 2-symmetric hypersurface ofR3 contained in asphere is given by the union of bands of small radius around the equator and two meridians of thecrossing under an angle ofπ/2.

    Let ϕ be a motion alongP from p. Let V be a vector subspace ofNpP . We say thatthe restriction ofthe motionϕ to V is parallel if DXNu = 0 for everyu ∈ V and everyX ∈ T P .

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 247

    d

    l to

    otionseicn

    at

    Lemma 6.2. Letϕ be a motion alongP fromp. LetV be the maximal vector subspace ofNpP satisfyingthat the restriction of the motionϕ to V is parallel. Letd be the dimension ofV . If the condition (6.2)holds for this vector spaceV , then

    (a) whend = 0, the equality holds in(6.1) if and only ifCp is contained in a geodesic sphere inMn−qλp .If, moreover,Cp is closed, then it is a sphere;

    (b) whend > 0 andCp is closed(and, whenλ > 0, dist(z,expp V ) <π

    2√

    λfor everyz ∈ Cp), the equality

    holds in (6.1) if and only ifCp is a revolution hypersurface aroundexpp V .

    Proof. After Lemma 6.1, it only remains to be proved that ifCp is a revolution hypersurface arounexpp V , then the equality in(6.1) holds, which we know is equivalent to(6.3). In order to see that(6.3)holds, letu ∈NpP −V be a unit vector; we can decomposeu = u1 + u2, with u1 ∈ V andu2 ∈ V ⊥, then

    〈DeaNu, ξ 〉 = |u2|〈DeaNu2/|u2|, ξ 〉.Following the same notation than in the proof ofLemma 6.1, for everyz = expp tu ∈ Cp we have that

    (6.11)TzU =〈γ ′

    (|w|)〉 ⊕ PsTf (z)V.Since Cp is a revolution hypersurface,Cp ∩ Mn−q−dλf (z) is a geodesic sphere, then it is orthogonaγ ′(s) ∈ TzU . Moreover,Tz(Cp ∩ Mn−q−dλf (z) ) ⊂ TzMn−q−dλf (z) is orthogonal toPsTf (z)V . Then it followsfrom (6.11)thatTz(Cp ∩ Mn−q−dλf (z) ) is orthogonal toTzU . But ξ̄ is orthogonal toTz(Cp ∩ Mn−q−dλf (z) ), thenξ̄ is tangent toTzU , andξ ∈ TpU . Therefore, if we decomposeξ = ξ1 + ξ2, with ξ1 ∈ V andξ2 ∈ V ⊥, wehave thatξ2 = cu2 for some constantc,

    〈DeaNu2/|u2|, ξ 〉 = 〈DeaNu2/|u2|, ξ2〉 = 0.If u ∈ V , it is obvious from the definition ofV thatDeaNu = 0. �

    In order to get a more geometric interpretation of the above lemmas, we shall describe the mfrom p alongP in a neighborhood ofp as immersions ofP in SO(n − q). For it, let us consider thopen setP [p] of P of all the pointsx ∈ P which can be joined top by a unique minimizing geodes(P [p] = P iff P is starlike with respect top, which includes the case whereP is a complete Riemanniamanifold without the cut points ofp). Let {ϕpx }x∈P [p] be the radial motion fromp alongP [p]. Given anymotion {ϕx}x∈P from p alongP , we know that the corresponding mapsϕp, ϕ :P [p] × NpP → NPare C∞, then, for everyξ ∈ NpP , the map fromP [p] to NpP defined byx �→ (ϕpx )−1 ◦ ϕx(ξ) isC∞. Moreover, sinceϕpx and ϕx are isometries,ϕ

    pp = ϕp = Id and P [p] is connected, we have th

    (ϕpx )

    −1 ◦ ϕx ∈ SO(NpP ) ≡ SO(n − q) for everyx ∈ P [p]. Then, to every motionϕ from p alongP wecan associate aC∞ map

    A :P [p] → SO(n − q), A(x) = (ϕpx )−1 ◦ ϕx.On a neighborhood ofp, we can composeA with the inverse ln= e−1 of the exponential map

    e :o(n − q) → SO(n − q) between the Lie algebrao(n − q) of SO(n − q) and SO(n − q). Then weget that a motionϕ has associated aC∞ map defined on a neighborhood ofp in P [p] by

    lnA :P [p] → o(n − q) x �→ lnA(x) :NpP → NpP.

  • 248 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    es

    s

    Every d-dimensional vector subspaceV ⊂ NpP defines an embedding ofSO(n − q − d) intoSO(n − q) as the group of rotations ofNpP which hasV as the set of fixed points, and it inducan embedding of the algebrao(n − q − d) as a subalgebra ofo(n − q), andV is the set of points inNpPwhich go to 0 by the action of all the elements ofo(n − q − d). From now on,o(n − q − d) will denoteany of these subalgebras.

    Given a motionϕ along P from p, and the associated mapA :P [p] → SO(n − q), we shall usdenote byP̃ [p] the maximal starlike open set containingp on which lnA is well defined. Letd bethe maximal natural number such that there is a subalgebrao(n − q − d) of o(n − q) satisfying thatlnA(P̃ [p]) ⊂ o(n − q − d). ThenTheorem 6.3. If lnA(P̃ [p]) is not contained in any hyperplane ofo(n − q − d) and the equality in(6.1)holds, then

    (a) whend = 0, Cp is contained in a geodesic sphere inMn−qλp . If, moreover,Cp is closed, then it is asphere;

    (b) whend > 0 and Cp is closed(and, whenλ > 0, dist(z,expp V ) <π

    2√

    λfor everyz ∈ Cp), Cp is a

    revolution hypersurface aroundexpp V , whereV is the set of fixed points ofNpP under the actionof SO(n − q − d).

    Proof. First, we shall rewrite condition(6.2)using lnA. Let ∂� = gradP �p. Now, let us observe that(ϕxi )

    −1D∂�Nu(xi) = (ϕxi )−1 ◦ ϕpxi ◦(ϕpxi

    )−1D∂�Nu(xi) = A(xi)−1

    (ϕpxi

    )−1D∂�Nu(xi).

    To write (ϕpxi )−1D∂�Nu(xi) in terms ofA, let eq+1, . . . , en be an orthonormal basis ofNpP , and letA

    j

    i (x)

    be the matrix ofA(x) in this basis. Ifu = ∑ni=q+1 uiei , thenϕx(u) = ϕpx

    (Ax(u)

    ) = n∑i,j=q+1

    Aj

    i uiϕpx ej =

    n∑i,j=q+1

    Aj

    i uiP

    p

    �p(x)(ej ),

    then

    D∂�Nu =n∑

    i,j=q+1

    ∂Aj

    i

    ∂�uiϕpx (ej ),

    and, taking the composition on the left withA(x)−1 ◦ (ϕpx )−1,(6.12)ϕ−1x D∂�Nu = A(x)−1

    ∂A(x)

    ∂�(u) = ∂ lnA(x)

    ∂�(u).

    Then, condition(6.2) is satisfied if the following condition(6.13)holds:

    There is a vector subspaceV of NpP of dimensiond satisfying thatfor every unit vectoru = Ny(p) ∈ NpP − V with y ∈ Cp, there arepointsxq+d+2, . . . , xn ∈ P̃ [p] such that the vectors(lnA)∗xi (∂�)(u)

    (6.13)are linearly independent and orthogonal toV.

    Now, we shall prove that, under the hypotheses of the lemma, condition(6.13)holds, then the thesiwill follow from Lemma 6.1. Since the elements ofV under the action ofo(n − q − d) go to zero, we

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 249

    by

    motion

    (i)om

    have that

    (6.14)lnA(x)v = 0 for everyv ∈ V and everyx ∈ P̃ [p].For everyu = Ny(p) ∈NpP − V , let us define

    k(u) = max{m; there arex1, . . . , xm; ∂ lnA

    ∂�(x1)(u), . . . ,

    ∂ lnA

    ∂�(xm)(u)

    are linearly independent and orthogonal toV

    }.

    From(6.14)and the definition ofk(u), we have that, for everyx ∈ P̃ [p], ∂ lnA∂�

    (x)(u) is in the vectorspace generated by∂ lnA

    ∂�(x1)(u), . . . ,

    ∂ lnA∂�

    (xk(u))(u). If (6.13) is not true, then there is au = Ny(p) ∈NpP − V such thatk(u) � n − q − d − 2. On the other hand, from the definition of̃P [p], the geodesicsof P starting fromp fill in all P̃ [p]. Given one of such geodesicsγ (�), its image lnA(γ (�))(u) is a curvein NpP through lnA(p)(u) and with tangent vector∂ lnA(γ (�))∂� (x)(u) in the vector space generated∂ lnA∂�

    (x1)(u), . . . ,∂ lnA∂�

    (xk(u))(u). Then, for thisu, the image of̃P [p] by the map lnA(·)u :x �→ lnA(x)uis contained inEk(u), whereEk(u) is a vector subspace ofNpP of dimensionk(u) � n − q − d − 2 andorthogonal toV .

    Now, let us write the matrices of the endomorphisms ino(n−q −d) using an orthonormal basis ofV ⊥(the complementary orthogonal toV in NpP ) of the form{ū, eq+d+2, . . . , en}, whereū is the unit vectorin the direction of the projectionu2 of u ontoV ⊥, and identifyo(n − q − d) with R(n−q−d)(n−q−d−1)/2 by

    0 −aq+d+1 q+d+2 . . . −aq+d+1 naq+d+1 q+d+2 0 . . . −aq+d+2 n

    ......

    . . ....

    aq+d+1 n aq+d+2 n . . . 0

    �→ (aq+d+1 q+d+2, aq+d+1 q+d+3, . . . , an−2 n, an−1 n).

    With this identification,

    lnA(x)(u) = |u2| lnA(x)(ū)= |u2|

    (0, lnA(x)q+d+1 q+d+2, . . . , lnA(x)q+d+1 n,0, . . . ,0

    )= |u2|πRn−q−d−1×{0} lnA(x) ∈ Ek(u),

    then lnA(P̃ [p]) ⊂ (πRn−q−d−1×{0})−1(Ek(u)), which is a vector subspace ofo(n − q − d) of dimension(n − q − d)(n − q − d − 1)/2 − (n − q − d − 1) + k(u) � (n − q − d)(n − q − d − 1)/2 − 1, incontradiction with the hypothesis.�

    There are three undesirable facts inTheorem 6.3: (i) it applies only to a neighborhood̃P [p] of p in P ;(ii) it left outside of its hypotheses many interesting motions, like, in many cases, the same radialfrom p, and (iii) it is only a necessary condition to have the equality in(6.1), and it will be desirableto have some characterization of the equality. The nextTheorem 6.4overcomes the undesirable factsand (ii), andTheorem 6.6overcomes (iii). To state these theorems we need to consider a motion frpas a motion from an arbitrary pointy.

  • 250 M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251

    ce

    at theion

    ry

    or

    in

    Given any motionϕ from p along P , and anyy ∈ P , ϕ defines a motionyϕ from y along P byyϕx = ϕx ◦ ϕ−1y . Let us denote byAy the map fromP [y] to SO(n − q) associated toyϕ as above. Letdybe now the maximal natural number such that lnAy(P̃ [y]) ⊂ o(n − q − dy), whereo(n − q − dy) is thesubalgebra of the skewsymmetric endomorphismso(n − q) of NyP which annihilates a vector subspaVy of NyP of dimensiondy .

    Theorem 6.4. For everyy ∈ P , let dy be the number defined above. Letd = min{dy; y ∈ P }. If there issomey ∈ P with dy = d satisfying thatlnAy(P̃ [y]) is not contained in any hyperplane ofo(n − q − d)and we have the equality in(6.1), then

    (a) whend = 0, Cp is contained in a geodesic sphere inMn−qλp . If, moreover,Cp is closed, then it is asphere;

    (b) whend > 0 andCp is closed(and, whenλ > 0, dist(z,expp ϕ−1y Vy) <

    π

    2√

    λfor everyz ∈ Cp), Cp is

    a revolution hypersurface aroundexpp ϕ−1y Vy .

    Proof. Just applyTheorem 6.3to the motionyϕ and recall thatCy andCp are isometric. �Remarks. When d = 0, it is possible to show, by standard arguments of generic geometry, thcondition ofTheorems 6.3 and 6.4are generic, that is, the family of motions satisfying this conditcontains an open and dense set in the space of motions with an appropriate topology.

    Whend > 0, the condition is generic among all the motions for which there is a subspaceV of NpPof dimensiond such that the motion restricted toV is parallel. These kind of motions exist on evesubmanifoldP having a subbundle ofNP of rankd with trivial D-holonomy.

    Theorems 6.3 and 6.4generalize Theorem 4.4 in[2] in two directions. On one side, it is allowed fthe dimensionq of P to be greater than 1. On the other side, even in dimensionq = 1 it characterizesthe revolution hypersurfaces as the only ones where we can get the minimum for volume(C) for motionswhich are parallel only restricted to some subspaceV , proving something that the reading of example[2, Remark 4.7]can easily suggest.

    Lemma 6.5. The restriction of a motionϕ alongP from p to a vector subspaceV of NpP is parallel ifand only if, for everyy ∈ P , the restriction ofyϕ to Vy = ϕy(V) is radial. In particular, a motionϕ isparallel if and only if, for everyy ∈ P , yϕ is radial.

    Proof. It is obvious that if the restriction ofϕ to V is parallel then the restriction ofyϕ to Vy is radial.Now, let us suppose thatyϕ restricted toVy is radial for everyy ∈ P . For everyx ∈ P , let v1, . . . , vq bea basis ofTxP , and consider the pointsyi = expx tivi ∈ P , 1� i � q. For eachu ∈ V, the normal vectorfield Nu :x �→ ϕx(u) satisfiesNu(x) = ϕx(u) = (yiϕ)x(ϕyi (u)) = Nyiϕyi (u)(x), whereNyiv :x �→ yiϕx(v) isthe unit normal vector field associated tov for the motionyi ϕ. Then

    DviNu = DviNyiϕyi (u) = 0,becauseyiϕx is theD-parallel transport along the geodesic ofP starting fromyi with tangent vector−viat x. This shows thatϕ is a parallel motion when restricted toV. �

  • M.C. Domingo-Juan, V. Miquel / Differential Geometry and its Applications 21 (2004) 229–251 251

    l

    ,

    e

    , Paris,

    ath. 128

    (1982)

    41–254.

    Theorem 6.6. Let Vy be as inTheorem6.4. If, for everyy ∈ P , Vy = ϕy(Vp) (thendy = dp) and there isa y ∈ P satisfying thatlnAy(P̃ [y]) is not contained in any hyperplane ofo(n − q − d), then

    (a) whend = 0, we have the equality in(6.1) if and only ifCp is contained in a geodesic sphere inMn−qλp .If, moreover,Cp is closed, then it is a sphere;

    (b) whend > 0 and Cp is closed(and, whenλ > 0, dist(z,expp V ) <π

    2√

    λfor everyz ∈ Cp), we have

    the equality in(6.1) if and only ifCp is a revolution hypersurface aroundexpp Vp.

    Proof. After Theorem 6.4, we have only to prove that ifCp is a revolution hypersurface around expp V ,thenV satisfies the hypothesis ofLemma 6.2. But this is a consequence ofLemma 6.5. �Corollary 6.7. If n − q = 2 andϕ is not a parallel motion, the equality holds in(6.1) if and only ifCp isan arc of a geodesic circle ofM2λp.

    Proof. This result will follow fromTheorem 6.6(a)if we show that, whenn − q = 2, not to be paralleimplies that there is ay ∈ P such that lnAy(P̃ [y]) is not contained in any hyperplane ofo(n − q). ByLemma 6.5, if ϕ is not parallel, there is ay ∈ P suchyϕ is not radial. ThereforeAy is not constant, andthen, it is not contained in an affine hyperplane ofo(2) which has dimension 1.�

    Acknowledgements

    We want to thank A.M. Naveira for directing our attention on theith fundamental forms and thRef. [13], and to F.J. Carreras for his help at different points.

    References

    [1] P. Buser, H. Karcher, Gromov’s Almost Flat Manifolds, in: Astérisque, vol. 81, Société Mathématique de France1981.

    [2] M.C. Domingo-Juan, X. Gual, V. Miquel, Pappus type theorems for hypersurfaces in a space form, Israel J. M(2002) 205–220.

    [3] H. Flanders, A further comment of Pappus, Amer. Math. Monthly 77 (1970) 965–968.[4] W. Goodman, G. Goodman, Generalizations of the theorems of Pappus, Amer. Math. Monthly 76 (1969) 355–366.[5] A. Gray, Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, Topology 21

    201–228.[6] A. Gray, Tubes, second ed., Birkhäuser, Basel, 2003.[7] A. Gray, V. Miquel, On Pappus-type theorems on the volume in space forms, Ann. Global Anal. Geom. 18 (2000) 2[8] E. Heintze, Extrinsic Upper Bounds forλ1, Math. Ann. 280 (1988) 389–402.[9] H. Karcher, Riemannian center of mass and Mollifier smoothing, Comm. PureAppl. Math.30 (1977) 509–541.

    [10] W.A. Poor, Differential Geometric Structures, McGraw-Hill, London, 1981.[11] L.E. Pursell, More generalizations of a theorem of Pappus, Amer. Math. Monthly 77 (1970) 961–965.[12] T. Sakai, Riemannian Geometry, American Mathematical Society, Providence, RI, 1996.[13] M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 4, Publish or Perish, Boston, MA, 1975.[14] H. Weyl, On the volume of tubes, Amer. J. Math. 61 (1939) 461–472.

    Pappus type theorems for motions along a submanifoldIntroductionMotion along a submanifold and tubes of nonspherical sectionA general formula to compute the volume of D in MnlambdaConsequences of Theorem 3.1 and a Weyl type formula for motionsA formula for volume(C) in MnlambdaWhen does volume(C) attain its lower bound?AcknowledgementsReferences