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An Introduction to Riemann-Finsler Geometry

verfasst von: D. Bao, S.-S. Chern, Z. Shen

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Professional Book Archive

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In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe?
It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one.

Inhaltsverzeichnis

Frontmatter

Finsler Manifolds and Their Curvature

Chapter 1. Finsler Manifolds and the Fundamentals of Minkowski Norms

Abstract

Finsler geometry has its genesis in integrals of the form

$$\int_a^b {F\left( {{x^1}, \ldots ,{x^n};\frac{{d{x^1}}}{{dt}}, \ldots ,\frac{{d{x^n}}}{{dt}}} \right)} dt$$

D. Bao, S.-S. Chern, Z. Shen

Chapter 2. The Chern Connection

Abstract

The Chern connection that we construct is a linear connection that acts on a distinguished vector bundle π*TM, sitting over the manifold TM \0 or SM. It is not a connection on the bundle TM over M. Nevertheless, it serves Finsler geometry in a manner that parallels what the Levi-Civita (Christoffel) connection does for Riemannian geometry. This connection is on equal footing with, but is different from, those due to Cartan, Berwald, and Hashiguchi (to name just a few).

D. Bao, S.-S. Chern, Z. Shen

Chapter 3. Curvature and Schur’s Lemma

Abstract

The curvature 2-forms of the Chern connection are

$$\boxed{\Omega _j^i: = dw_j^i - w_j^k \wedge w_k^i}$$

(3.1.1)

D. Bao, S.-S. Chern, Z. Shen

Chapter 4. Finsler Surfaces and a Generalized Gauss-Bonnet Theorem

Abstract

So far, our treatment has emphasized the use of natural coordinates. At the beginning of Chapter 2, we stated our policy that in important computations, we only use objects which are invariant under positive rescaling in y. Consequently, our treatment using natural coordinates on TM \0 can be regarded as occurring on the (projective) sphere bundle SM, in the context of homogeneous coordinates.

D. Bao, S.-S. Chern, Z. Shen

Calculus of Variations and Comparison Theorems

Chapter 5. Variations of Arc Length, Jacobi Fields, the Effect of Curvature

Abstract

In this section, we use the method of differential forms to describe the first variation. There is another approach which uses vector fields and covariant differentiation. That is explored in a series of guided exercises at the end of 5.2. (Those exercises involve the second variation as well.) A systematic self-contained account can also be found in [BC1].

D. Bao, S.-S. Chern, Z. Shen

Chapter 6. The Gauss Lemma and the Hopf-Rinow Theorem

Abstract

Fix x ∈ M. In T _x M, we define the tangent spheres

$${S_x}(r): = \{ y \in {T_x}M:F(x,y) = r\} $$

(6.1.1)

and open tangent balls

$${B_x}(r): = \{ y \in {T_x}M:F(x,y) = r\} $$

(6.1.2)

of radii r. The exponential map exp_x is a local diffeomorphism at the origin of T _x M because its derivative there is the identity; see §5.3. Thus, for r small enough, not only does exp_x [S_x (r)] makes sense, it is also diffeomorphic to S_x (r). The image set

$${\exp _x}[{S_x}(r)]$$

is called a geodesic sphere in M centered at x. We later show why it can be said to have radius equal to r.

D. Bao, S.-S. Chern, Z. Shen

Chapter 7. The Index Form and the Bonnet—Myers Theorem

Abstract

Let (M, F) be a Finsler manifold, where F is C∞ on TM \0 and is positively (but perhaps not absolutely) homogeneous of degree 1. Fix T ∈ T _p M. Consider the constant speed geodesic σ(t) = exp_p (tT), 0 ⩽ t ⩽ r that emanates from p = σ(0) and terminates at q = σ(r). If there is no confusion, label its velocity field by T also. Let D _T denote covariant differentiation along σ, with reference vector T. This concept was introduced in the Exercise portion of 5.2.

D. Bao, S.-S. Chern, Z. Shen

Chapter 8. The Cut and Conjugate Loci, and Synge’s Theorem

Abstract

In this chapter, the following assumptions are made throughout:

•The Finsler structure F is positively (but perhaps not absolutely) homogeneous of degree one. Consequently, the associated metric distance function d may not be symmetric.
• The Finsler manifold (M, F) is forward geodesically com plete. That is, all geodesics, parametrized to have constant Finslerian speed, are indefinitely forward extendible. By the Hopf-Rinow theorem, this is equivalent to saying that all forward Cauchy se quences are convergent. See Theorem 6.6.1 and §6.2D. Also, accord ing to Exercise 6.6.4, the completeness hypothesis is automatically satisfied whenever M is compact.
• Unless we specify otherwise, all geodesics are parametrized to have unit Finslerian speed.

D. Bao, S.-S. Chern, Z. Shen

Chapter 9. The Cartan—Hadamard Theorem and Rauch’s First Theorem

Abstract

In §5.5, we estimated the growth of certain Jacobi fields using the first few terms of a power series. That was valid only for a short time interval. In the present section, we use a more delicate approach—known as a comparison argument. The resulting estimate holds for long time intervals.

D. Bao, S.-S. Chern, Z. Shen

Special Finsler Spaces over the Reals

Chapter 10. Berwald Spaces and Szabó’s Theorem for Berwald Surfaces

Abstract

In this chapter, we study Berwald spaces in some detail. Here are several reasons why such spaces are so important. These reasons are elaborated upon as the chapter unfolds.

D. Bao, S.-S. Chern, Z. Shen

Chapter 11. Randers Spaces and an Elegant Theorem

Abstract

In 1941, G. Randers [Ra] studied a very interesting type of Finsler structures. These are called Randers metrics, and we first encountered them in §1.3. Randers metrics are important for six reasons.

D. Bao, S.-S. Chern, Z. Shen

Chapter 12. Constant Flag Curvature Spaces and Akbar-Zadeh’s Theorem

Abstract

In §3.9, we encountered the flag curvature. As the name suggests, this quantity (denoted K) involves a location x ϵ M, a flagpole ℓ:= with y ϵ T ^x M, and a transverse edge V ϵ T ^x M. The precise formula is quite elegantly given by (3.9.3):

$$K(\ell ,V): = \frac{{{V^i}({\ell ^j}{R_{jikl}}{\ell ^l}){V^k}}}{{g(\ell ,\ell )g(V,V) - {{[g(\ell ,V)]}^2}}} = \frac{{{V^i}{R_{ik}}{V^k}}}{{g(V,V) - {{[g(\ell ,V)]}^2}}}$$

D. Bao, S.-S. Chern, Z. Shen

Chapter 13. Riemannian Manifolds and Two of Hopf’s Theorems

Abstract

A Riemannian metric g on a manifold M is a family of inner products {g x}x∈M such that the quantities

$${g_{ij}}(x): = g\left( {\frac{\partial }{{\partial {x^i}}},\frac{\partial }{{\partial {x^i}}}} \right)$$

are smooth in local coordinates. The Finsler function F(x, y) of a Riemannian manifold has the characteristic structure

$$F(x,y) = \sqrt {{g_{ij(x)}}{y^i}{y^j}} $$

D. Bao, S.-S. Chern, Z. Shen

Chapter 14. Minkowski Spaces, the Theorems of Deicke and Brickell

Abstract

Let y y ↦ F(y) be a Minkowski norm on ⁿ. It is nonnegative and has the following defining properties.

D. Bao, S.-S. Chern, Z. Shen

Backmatter

Titel: An Introduction to Riemann-Finsler Geometry
verfasst von: D. Bao
S.-S. Chern
Z. Shen
Verlag: Springer New York
Electronic ISBN: 978-1-4612-1268-3
Print ISBN: 978-1-4612-7070-6
DOI: https://doi.org/10.1007/978-1-4612-1268-3

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Finsler Manifolds and Their Curvature

Chapter 1. Finsler Manifolds and the Fundamentals of Minkowski Norms

Chapter 2. The Chern Connection

Chapter 3. Curvature and Schur’s Lemma

Chapter 4. Finsler Surfaces and a Generalized Gauss-Bonnet Theorem

Calculus of Variations and Comparison Theorems

Chapter 5. Variations of Arc Length, Jacobi Fields, the Effect of Curvature

Chapter 6. The Gauss Lemma and the Hopf-Rinow Theorem

Chapter 7. The Index Form and the Bonnet—Myers Theorem

Chapter 8. The Cut and Conjugate Loci, and Synge’s Theorem

Chapter 9. The Cartan—Hadamard Theorem and Rauch’s First Theorem

Special Finsler Spaces over the Reals

Chapter 10. Berwald Spaces and Szabó’s Theorem for Berwald Surfaces

Chapter 11. Randers Spaces and an Elegant Theorem

Chapter 12. Constant Flag Curvature Spaces and Akbar-Zadeh’s Theorem

Chapter 13. Riemannian Manifolds and Two of Hopf’s Theorems

Chapter 14. Minkowski Spaces, the Theorems of Deicke and Brickell

Backmatter