Handbook

of

Differential

Geometry

•

VOLUME II

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Handbook

of

Differential

Geometry

•

VOLUME II

Editors

Franki J.E. Dillen

Leopold C.A. Verstraelen

Katholieke Universiteit Leuven

Department of Mathematics

Leuven, Belgium

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First edition 2006

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Dedication

In memory of S.S. Chern and T. Willmore

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Preface

“Our goal with the volumes which together will constitute the “Handbook of Differential Geometry” is to give a rather complete survey of the field of differential geometry.”

Thus reads the opening sentence of the “Handbook of Differential Geometry, Volume I”,

and only the presence of the word “rather” saves this goal from being an obvious mission impossible. Let us recall the contents of this Volume I: Differential geometry of webs

(M.A. Akivis and V.V. Goldberg), Spaces of metrics and curvature functionals (D.E. Blair),

Riemannian submanifolds (B.-Y. Chen), Einstein metrics in dimension four (A. Derdzinski), The Atiyah–Singer index theorem (P.B. Gilkey), Survey of isospectral manifolds

(C.S. Gordon), Submanifolds with parallel fundamental form (Ü. Lumiste), Sphere theorems (K. Shiohama), Affine differential geometry (U. Simon), A survey on isoparametric

hypersurfaces and their generalizations (G. Thorbergsson), Curves (T. Willmore); with

introduction by S.S. Chern.

As in Volume I, we allowed the authors in this Volume II as much freedom as possible

concerning style and contents. We are confident that the reader will appreciate this pragmatic point of view. Some contributions will emphasize the basics; some will emphasize

the classical results; others the recent developments. Needless to say all authors have spent

a lot of time and energy in describing their topic, which we appreciate enormously.

The contributions to this Volume II are: Some problems on Finsler geometry (J.C. Álvarez Paiva), Foliations (R. Barre and A. El Kacimi), Symplectic geometry (A. Cannas da Silva), Metric Riemannian geometry (K. Fukaya), Contact geometry (H. Geiges),

Complex differential geometry (I. Mihai), Compendium on the geometry of Lagrange

spaces (R. Miron), Certain actual topics on modern Lorentzian geometry (F.J. Palomo

and A. Romero).

Obviously the whole field of differential geometry is not yet covered in the two volumes

of this “Handbook of Differential Geometry”. Some of the authors explicitly mention topics that should have been covered, but are not for practical reasons; but also other topics

are not (yet) treated sufficiently or not treated at all.

Recently Professors Chern and Willmore passed away. Both had a great impact on the

development of contemporary geometry and were genuine sources of inspiration, guidance

and support for many generations of mathematicians through their books and articles, their

fantastic lectures and their warm and truly concerned personal contacts. Together with

all authors we gratefully dedicate this book to the memories of Professor S.S. Chern and

Professor T.J. Willmore.

Franki Dillen and Leopold Verstraelen

vii

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List of Contributors

Álvarez Paiva, J.C., Polytechnic University, Brooklyn, NY (Ch. 1).

Barre, R., Université de Valenciennes, Valenciennes (Ch. 2).

Cannas da Silva, A., Instituto Superior Técnico, Lisboa (Ch. 3).

El Kacimi Alaoui, A., Université de Valenciennes, Valenciennes (Ch. 2).

Fukaya, K., Kyoto University, Kyoto (Ch. 4).

Geiges, H., Universität zu Köln, Köln (Ch. 5).

Mihai, I., University of Bucharest, Bucharest (Ch. 6).

Miron, R., “Al.I. Cuza” University Iasi, Iasi (Ch. 7).

Palomo, F.J., Universidad de Málaga, Málaga (Ch. 8).

Romero, A., Universidad de Granada, Granada (Ch. 8).

ix

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Contents

Preface

List of Contributors

Contents of Volume I

1. Some problems on Finsler geometry

J.C. Álvarez Paiva

2. Foliations

R. Barre and A. El Kacimi Alaoui

3. Symplectic geometry

A. Cannas da Silva

4. Metric Riemannian geometry

K. Fukaya

5. Contact geometry

H. Geiges

6. Complex differential geometry

I. Mihai

7. Compendium on the geometry of Lagrange spaces

R. Miron

8. Certain actual topics on modern Lorentzian geometry

F.J. Palomo and A. Romero

Author Index

Subject Index

vii

ix

xiii

1

35

79

189

315

383

437

513

547

555

xi

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Contents of Volume I

Preface

Introduction

List of Contributors

1. Differential geometry of webs

M.A. Akivis and V.V. Goldberg

2. Spaces of metrics and curvature functionals

D.E. Blair

3. Riemannian submanifolds

B.-Y. Chen

4. Einstein metrics in dimension four

A. Derdzinski

5. The Atiyah–Singer index theorem

P.B. Gilkey

6. Survey of isospectral manifolds

C.S. Gordon

7. Submanifolds with parallel fundamental form

Ü. Lumiste

8. Sphere theorems

K. Shiohama

9. Affine differential geometry

U. Simon

10. A survey on isoparametric hypersurfaces and their generalizations

G. Thorbergsson

11. Curves

T. Willmore

Author Index

Subject Index

v

vii

ix

1

153

187

419

709

747

779

865

905

963

997

1025

1037

xiii

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CHAPTER 1

Some Problems on Finsler Geometry∗

J.C. Álvarez Paiva∗∗

Department of Mathematics, Polytechnic University, Six MetroTech Center, Brooklyn, NY 11201, USA

E-mail: jalvarez@duke.poly.edu

We do like intuitive geometric arguments and uncovering

simple geometric reasons underlying seemingly recondite facts.

H. Busemann

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . .

2. Preliminaries . . . . . . . . . . . . . . . . . . . .

2.1. The Hamiltonian point of view . . . . . . . .

2.2. The Riemannian point of view . . . . . . . .

2.3. Isometries and isometric embeddings . . . .

2.4. Isometric submersions . . . . . . . . . . . .

3. Volume and area in Finsler spaces . . . . . . . . .

4. Unit spheres in Minkowski spaces . . . . . . . . .

5. Symplectic equivalence of Finsler manifolds . . .

5.1. Equivalence of unit co-disc bundles . . . . .

5.2. Equivalence of unit co-sphere bundles . . . .

5.3. Manifolds of geodesics . . . . . . . . . . . .

6. Around Hilbert’s fourth problem . . . . . . . . . .

7. Closed geodesics . . . . . . . . . . . . . . . . . .

8. Differential invariants of Finsler surfaces . . . . .

8.1. Convex geometry and the invariant I . . . .

8.2. The invariant J . . . . . . . . . . . . . . . .

8.3. Curvature of Finsler surfaces . . . . . . . . .

8.4. Cartan’s structure equations . . . . . . . . .

8.5. Applications of Cartan’s structure equations

References . . . . . . . . . . . . . . . . . . . . . . .

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31

* This work was partially funded by a crédit aux chercheurs from the FNRS, Belgium.

** I warmly thank G. Berck, D. Burago, C. Durán, E. Fernandes, P. Foulon, I.M. Gelfand, R. Schneider, Z. Shen

and S. Tabachnikov for some very useful conversations and/or criticisms of earlier versions of this work.

HANDBOOK OF DIFFERENTIAL GEOMETRY, VOL. II

Edited by F.J.E. Dillen and L.C.A. Verstraelen

© 2006 Elsevier B.V. All rights reserved

1

2

J.C. Álvarez Paiva

Abstract

This chapter is an unorthodox survey of Finsler geometry presenting both results and open

problems. It aims to show that recent progress in convex geometry, the calculus of variations,

symplectic geometry, and integral geometry can be powerful tools in the study of Finsler

manifolds; and that Finsler geometry can prove useful in solving some of the open problems

in these fields.

Some problems on Finsler geometry

3

1. Introduction

Finsler manifolds, manifolds whose tangent spaces carry a norm that varies smoothly with

the base point, were born prematurely in 1854 together with their Riemannian counterparts

in Riemann’s ground-breaking Habilitationsvortrag. I say prematurely because in 1854

Minkowski’s work on normed spaces and convex bodies (see [69]) was still forty three

years away, and thus not even the infinitesimal geometry on which Finsler manifolds are

based was understood or appreciated at the time. Apparently, Riemann did not know what

to make of these ‘more general class’ of manifolds whose element of arclength does not

originate from a scalar product and, fatefully, put in a bad word for them [44]:

Investigation of this more general class would actually require no essential different principles,

but it would be rather time-consuming and throw relatively little new light on the study of Space,

especially since the results cannot be expressed geometrically.

Given the awe with which we rightfully regard Riemann’s achievements and uncanny

geometrical intuition, it is tempting to take the above quotation out of historical context

and to dismiss Finsler geometry altogether. But, if we think of the great advances in convex

geometry, the calculus of variations, integral geometry, the theory of metric spaces, and

symplectic geometry that have taken place since 1854, then we may be moved to reassess

Riemann’s statement and to consider applying these new tools to develop the subject in a

way that Riemann could not have foreseen.

The paper includes eighteen simply-stated open problems, as well as a survey of the

more elementary and geometric chapters of Finsler geometry. It presents a detailed discussion of the Holmes–Thompson volume and its role in integral geometry and geometric

inequalities, thus complementing the survey by Álvarez and Thompson [16]. The other

highlights of the paper are its presentation of Hilbert’s fourth problem and its elementary

approach to the differential invariants of Finsler surfaces. These are mostly based on the papers [15,12,13] with I.M. Gelfand, M. Smirnov and E. Fernandes, as well as on the lecture

notes [10] written jointly with C. Durán.

In view of the often-made criticisms of Finsler geometry—very few concrete and interesting examples, very few non-Riemannian theorems of real geometric content, and too

many subindices—I have tried to include as many concrete examples, simply-stated results, and geometric constructions as possible. In this way, many of the jewels, so to speak,

of Finsler geometry find their way into the following pages.

As anyone writing a survey paper, I have had to make some choices. In matters of taste,

I have consistently preferred the concrete to the abstract, the elementary to the advanced,

the C ∞ to the C k , and the global to the local. I have stayed clear of Riemann–Finsler

geometry and Finsler connections because the book [23] of Bao, Chern and Shen covers

the subject in depth as do the lecture notes of Abate and Patrizio [1]. Because of my ignorance of the subject, I have not touched on complex Finsler geometry (see [1] also for this

topic) and, despite their undeniable interest and importance, non-reversible Finsler metrics

are barely mentioned. Another important topic that is not covered in this survey is Busemann’s G-spaces. This approach, which consists in abstracting the properties of geodesics

on Finsler manifolds, is one of the most powerful in Finsler geometry, but it is impossible

to outdo Busemann’s own exposition in [37,38].

4

J.C. Álvarez Paiva

The reader wishing to have a broader view of the activity in Finsler geometry should

read the short surveys of Chern [44] and Busemann [35]. The book [20] contains survey

articles on several topics including non-reversible Finsler metrics, the proceedings [22]

contains many open problems and a vista of various approaches to Finsler geometry. The

book [84] contains a beautiful exposition of the convex-geometric aspects of the Holmes–

Thompson volume as well as most of the convex geometry necessary for the study of

Finsler manifolds. The lecture notes [10] are similar in spirit to the present paper. The

thesis of Egloff [46] is a good place to learn about the beautiful results of Egloff [47,48]

and Foulon [50–52] on the geometry and dynamics of Finsler manifolds with non-positive

and negative curvature. Finally, I wholeheartedly recommend looking at the papers [42]

and [3] before plunging into other papers where Finsler connections are treated.

The reader can find many of the preprints cited in this paper, along with other

works on the interaction between convex, integral, metric, and symplectic geometry, in

http://www.math.poly.edu/research/finsler.

2. Preliminaries

If (V , · ) is a real, finite-dimensional normed space, we define the length of a smooth

curve γ : [a, b] → V by the formula

b

length of γ :=

γ˙ (t) dt.

a

A smooth submanifold N ⊂ V inherits a metric from the norm: if x and y are two points

on N , define their distance as the infimum of the lengths of all smooth curves on N joining

x and y. Notice that in order to define the metric on N it suffices to know the restriction of

the norm to each tangent space. This motivates the following heuristic definition: A Finsler

manifold is a manifold together with the choice of a norm on each tangent space. The

precise definition requires us to restrict the class of norms to those where the unit sphere

is smooth and quadratically convex (i.e., it has positive principal curvatures for some (and

therefore any) Euclidean structure on V ). The intrinsic definition of these norms is as

follows:

Let V be a vector space and let ϕ : V → [0, ∞) be a norm that is smooth outside the

origin. Set L := ϕ 2 /2 and consider the exterior derivative of L, dL, as a map from V

minus the origin to V ∗ minus the origin. The norm ϕ is said to be a Minkowski norm if dL

is a diffeomorphism.

For any non-zero vector v ∈ V , the differential

D(dL)(v) : Tv V → TdL(v) V ∗

is an invertible linear map. In fact, using the natural identification of Tv V with V ,

and TdL(v) V ∗ with V ∗ , we can think of gϕ (v) := D(dL)(v) as a (symmetric) bilinear form

on V :

gϕ (v)(w1 , w2 ) := D(dL)(v)(w1 ) (w2 ).

Some problems on Finsler geometry

5

The norm ϕ is a Minkowski norm if and only if gϕ is positive definite. When the vector v

belongs to the unit sphere, we will denote gϕ (v) as the osculating Euclidean structure at v

and the ellipsoid

Ev := w ∈ V : gϕ (v)(w, w) = 1

as the osculating ellipsoid at v.

D EFINITION 2.1. A Finsler metric on a manifold M is a continuous function defined on

its tangent bundle with the property that it is smooth away from the zero section and its

restriction to each tangent space is a Minkowski norm.

Some examples of Finsler manifolds are submanifolds of Minkowski spaces and flat tori

obtained as quotients of Minkowski spaces.

If γ : [a, b] → M is a smooth curve on a Finsler manifold (M, ϕ), then the quantity

b

length of γ :=

ϕ γ˙ (t) dt

(1)

a

is independent of the parameterization. Using this definition of length we define a metric

on M by letting the distance between two points x, y ∈ M to be the infimum of the lengths

of all smooth curves joining x and y. Finsler manifolds are length spaces: the length of a

curve γ defined by the integral in (1) equals the metric length of the curve given by

k−1

dist γ (ti ), γ (ti+1 ) : a = t0 < · · · < tk = b is a partition of [a, b] .

sup

i=0

The condition that the norms in each tangent space be Minkowski norms is necessary

for the study of the geodesics. Namely, we want these to be solutions of a second-order

differential equation on M.

2.1. The Hamiltonian point of view

If (V , ϕ) is a normed space, then the dual vector space V ∗ inherits a natural norm defined

by the equation

ϕ ∗ (ξ ) := sup ξ (v) : ϕ(v)

1 .

A related construction on Minkowski spaces is the Legendre transform which assigns to

a non-zero vector v ∈ V the covector ϕ(v) dϕ(v) = gϕ (v)(v, ·). It is easy to check that if

v belongs to the unit sphere S ⊂ (V , ϕ), then the Legendre transform of v is the unique

covector ξ such that ξ = 1 is the hyperplane tangent to S at the point v. This implies that

the image of S under the Legendre transform is the unit sphere in (V ∗ , ϕ ∗ ).

6

J.C. Álvarez Paiva

Let (M, ϕ) be a Finsler manifold and for each point m ∈ M let ϕm denote the Minkowski

∗ ) is the dual of the normed space (T M, ϕ ), then the function

norm on Tm M. If (Tm∗ M, ϕm

m

m

H : T ∗M → R

∗ (p ) is a Hamiltonian whose energy surfaces are fiber-wise condefined by H (pm ) := ϕm

m

vex. Applying the Legendre transform on each tangent space of M defines a diffeomorphism L : T M \ 0 → T ∗ M \ 0 with the property that H ◦ L = ϕ.

By passing from a Finsler metric to its associated Hamiltonian, we gain access to the

techniques of Hamiltonian mechanics and symplectic geometry. Below, we recall some of

the basic definitions and constructions. For more information see [18] and [2].

D EFINITION 2.2. Let π : T ∗ M → M be the standard projection and let Dπ : T (T ∗ M) →

T M be its differential. The canonical 1-form α on T ∗ M is defined by the equation

α(vpm ) = pm (Dπ(vpm )), where pm ∈ Tm∗ M and vpm ∈ Tpm (T ∗ M). The symplectic 2-form

is defined as ω := −dα.

The form ω is non-degenerate: at each point pm ∈ T ∗ M, the map vpm → ωpm (vpm , ·) is

an isomorphism from Tpm (T ∗ M) to Tp∗m (T ∗ M). We can use this isomorphism to pass from

1-forms on T ∗ M to vector fields on T ∗ M.

D EFINITION 2.3. Let H : T ∗ M → R be a smooth function. The Hamiltonian vector field

of H , XH , is defined by the equality dH = ω(XH , ·).

As an easy consequence of the definition, we have that H is constant along the integral

curves of the Hamiltonian vector field XH , and that the symplectic form is invariant under

the flow of XH .

Because of this result, it is usual to disregard the function H in favor of the unit co∗ M := H −1 (1). If α is the canonical 1-form on T ∗ M, then its restriction

sphere bundle SH

∗ M, which we denote by α , is a contact form (i.e., the

to the unit co-sphere bundle SH

H

n−1

never vanishes). Using αH , we can define the restriction

top-order form αH ∧ (dαH )

of the Hamiltonian vector field XH without any reference to the function H :

∗ M is defined by the equations

D EFINITION 2.4. The Reeb vector field XH on SH

αH (XH ) = 1,

dαH (XH , ·) = 0.

The projection to M of the integral curves of this vector field are geodesics parameterized with unit speed. Conversely, if γ is a geodesic on M parameterized with unit speed,

then the Legendre transform L maps the velocity curve γ˙ to an orbit of the Reeb vector

field. We remark that if γ is any smooth curve on M parameterized with unit speed, then

length of γ =

L◦γ

αH .

(2)

Some problems on Finsler geometry

7

Note that the Finsler manifold (M, ϕ) is geodesically complete (or metrically complete,

since it is easy to verify that the Hopf–Rinow theorem extends to the Finsler setting) if and

only if the Reeb vector field defines a flow.

Let us finish this section by remarking that the non-degeneracy of the symplectic form ω

on T ∗ M is equivalent to the fact that ωn , n = dim(M), is a volume form. This remark will

provide us with a natural way to define the volume of a Finsler manifold.

2.2. The Riemannian point of view

Finsler manifolds can also be studied from the point of view of Riemannian manifolds and

bundles. Indeed, to every unit vector vm ∈ Tm M we may associate the inner product gϕ (v).

In this way, we can define a Riemannian structure on the pullback of the tangent bundle of

M to the unit tangent bundle of M. This construction underlies many of the definitions of

connections associated to Finsler manifolds.

A variation on this theme is to take a nowhere zero vector field X defined on an open subset O ⊂ M and to associate to it the Riemannian metric on O defined by m → gϕ (X(m)).

This construction has been used by Shen (see [79] and Section 8) to give a simple description of the Finsler curvature.

2.3. Isometries and isometric embeddings

The definitions of isometry and isometric embedding between Finsler manifolds (M, ϕM )

and (N, ϕN ) are the same as for Riemannian manifolds. Namely, an isometry (respectively

isometric embedding) is a diffeomorphism (respectively embedding) f : M → N such that

f ∗ ϕN = ϕM . Unlike Riemannian manifolds, two Finsler manifolds can fail to be isometric

because of what happens at a single tangent space. For example, if at a point m ∈ M the

indicatrix

Sm M := vm ∈ Tm M: ϕM (vm ) = 1

is an ellipsoid while none of the indicatrices of N are ellipsoids, then M and N are not

isometric. This remark points at the important role played by the centro-affine geometry of

convex hypersurfaces in Finsler geometry.

In [32] Burago and Ivanov showed that any compact Finsler manifold admits an isometric embedding into a finite-dimensional normed space. It is likely, but unproved, that

the norm can be chosen to be a Minkowski norm. They also give examples of noncompact Finsler manifolds that cannot be isometrically embedded in any finite-dimensional

normed space. Examples of Finsler manifolds that cannot be isometrically embedded in

any Minkowski space have been given by Shen (see [79]) and by Álvarez and Durán (see

Section 8.5).

However, new types of embedding problems arise in Finsler geometry. For example,

while it is known that every two-dimensional normed space is isometric to a subspace

of L1 ([0, 1]) (see, for example, [29]), it is not clear whether the following problem has an

affirmative answer.

8

J.C. Álvarez Paiva

P ROBLEM 1. Does every two-dimensional Finsler manifold admit an isometric embedding

into the Banach space L1 ([0, 1])?

The relation between intrinsic and extrinsic geometric properties of submanifolds of

normed space is not understood. For example, the following problem from [84] is open.

P ROBLEM 2 (Thompson). Let X and Y be two normed spaces of dimension n, n > 2,

such that its unit spheres are isometric as Finsler manifolds. Does it follow that X and Y

are isometric as normed spaces?

2.4. Isometric submersions

In Riemannian geometry isometric submersions are used to construct examples of Riemannian manifolds while keeping some control on their geodesics and curvature. The

Finslerian generalization of this construction is simple, but perhaps not as well known

as it should be. What follows is taken from [11].

D EFINITION 2.5. A surjective linear map π : X → Y between two normed spaces is said

to be an isometric submersion if the image of the closed unit ball on X under the map π

equals the closed unit ball on Y .

Clearly, if x is any vector in X, then π(x) Y

x X . The vectors for which the equality

holds are called horizontal vectors and form the horizontal cone in X. Notice that if the

unit sphere in X is smooth, its intersection with the horizontal cone is the singular set of

the restriction of the map π to the unit sphere. This description makes it easy to grasp that,

unlike the case where the spaces is Euclidean, the horizontal cone is rarely a subspace.

D EFINITION 2.6. A submersion ρ : M → N between Finsler manifolds is said to be isometric if for every point m ∈ M the differential

Dm ρ : Tm M → Tρ(m) N

is an isometric submersion of normed spaces.

More generally, Berestovskii has defined in [24] isometric submersions, or submetries of

metric spaces, as maps that send metric balls to metric balls of the same radius. When the

metric spaces are Finsler manifolds both notions of isometric submersion agree. However,

in the particular case of Finsler manifolds we can also speak of horizontal lifts.

D EFINITION 2.7. Let ρ : M → N be an isometric submersion. An immersed curve

γ : [a, b] → M is said to be horizontal if for every t ∈ (a, b) the velocity vector γ˙ (t) belongs to the horizontal cone in Tγ (t) M. A curve γ : [a, b] → M is said to be a horizontal

lift of an immersed curve σ : [a, b] → N if γ is horizontal and ρ ◦ γ = σ .

Some problems on Finsler geometry

9

The following result is an easy consequence of the definitions and of the basic properties

of geodesics in a Finsler manifold.

T HEOREM 2.8 (Álvarez and Durán [11]). An immersed curve on N is a geodesic if and

only if any of its horizontal lifts is a geodesic on M. In particular, the geodesics of N are

precisely the projections of horizontal geodesics on M.

As we shall see in Section 6, this theorem is useful in constructing interesting examples

of Finsler metrics on complex and quaternionic projective spaces.

3. Volume and area in Finsler spaces

The theory of volume and area in normed spaces has long been a major driving force in

convex geometry. For example, the Blaschke–Santaló inequality, the Mahler conjecture, the

Busemann–Petty problems, the Shephard problem, and the numerous works of Busemann,

Ewald and Shephard on the notions of convexity on Grassmannians originated from or

have applications to the study of volumes and areas in normed spaces (see [16]).

Defining a volume on a finite-dimensional normed space seems easy: a natural volume

should be invariant under translations, positive on open sets, and finite on compact sets. By

Haar’s theorem, such a volume must be a multiple of the Lebesgue measure. However, the

choice of this multiple is crucial. To understand this, suppose we have already decided on

how to assign those constants to two-dimensional normed spaces. We can now define the

area of a two-dimensional polyhedral surface embedded in a 3-dimensional normed space

as the sum of the areas of its faces with their induced norms. Making a different choice of

constants leads to a completely different way of measuring the area of polyhedral surfaces.

The guiding principle in defining volumes on normed spaces is that the choice of a

volume for every k-dimensional normed space leads to the definition of the k-volume integrand in all higher-dimensional normed spaces. Requiring even mild conditions on these

area integrands, such as that regions in hyperplanes be area-minimizing, severely restricts

our choices for a definition of volume. In fact, in the literature one can only find three

reasonable choices of volume on normed spaces: the Busemann definition, the Holmes–

Thompson definition, and the Benson definition (also known as Gromov’s mass∗ [56]).

Since the Busemann volume of a Finsler manifold coincides with its Hausdorff measure

as a metric space, at first sight it seems the most natural and geometric definition. However,

the Holmes–Thompson definition, with its ties to Brunn–Minkowski theory, integral, and

symplectic geometry, is rapidly becoming the definition of choice.

D EFINITION 3.1. The Holmes–Thompson volume of an n-dimensional Finsler manifold (M, ϕ), voln (M, ϕ), is the symplectic volume of its unit co-disc bundle divided by

the volume of the Euclidean n-dimensional unit ball. The k-volume of a k-dimensional

submanifold is the volume of the submanifold with its induced Finsler metric.

10

J.C. Álvarez Paiva

Note that, using the notation of the previous section, we have that

voln (M) =

1

n n!

∗M

SH

αH ∧ (dαH )n−1 ,

(3)

where n is the volume of the Euclidean unit ball of dimension n.

Using the Blaschke–Santaló inequality, Durán has remarked in [45] that the Hausdorff

measure of a Finsler manifold is no less than its Holmes–Thompson volume. Equality holds

if and only if the metric is Riemannian.

One of the basic problems about a given definition of volume on normed and Finsler

spaces is determining the convexity or ellipticity properties of the k-volume integrands.

Perhaps the most enticing and difficult question of the kind is the following problem of

Busemann:

P ROBLEM 3. Let P be a compact polyhedron of dimension k in a normed space. Is the

k-volume of any given face less than or equal to the sum of the k-volumes of the remaining

faces?

In the case of polyhedra of codimension one the answer to this question is affirmative for

both the Hausdorff measure (Busemann [34]) and the Holmes–Thompson volume (Holmes

and Thompson [62]).

For the Hausdorff measure no other results of this kind are known. For the Holmes–

Thompson volume the answer to Busemann’s question is known to be affirmative if the

normed space is a subspace of L1 ([0, 1]):

T HEOREM 3.2 (Busemann et al. [40]). Let P be a compact polyhedron of dimension k in

the Banach space L1 ([0, 1]). The (Holmes–Thompson) k-volume of any given face is less

than or equal to the sum of the k-volumes of the remaining faces.

The latest progress on the question of minimality of flats in normed spaces is the following result of Burago and Ivanov:

T HEOREM 3.3 (Burago and Ivanov [33]). Let P be a compact 2-dimensional polyhedron in a normed space. If P is homeomorphic to a sphere, then the (Holmes–Thompson)

2-volume of any given face is less than or equal to the sum of the 2-volumes of the remaining faces.

In [64], S. Ivanov shows that this result can be extended to a theorem that is new even in

the Riemannian case:

T HEOREM 3.4 (Ivanov [64]). Let ϕ be a Finsler metric on the closed two-dimensional disc

D such that every two points on D are joined by a unique geodesic. If ψ is another Finsler

metric on D such that the distance induced by ψ on the boundary ∂D of D is greater than

or equal to the distance induced by ϕ on ∂D, then the Holmes–Thompson volume of (D, ϕ)

does not exceed that of (D, ψ).

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