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Finsler Geometry Is Just Riemannian Geometry Without the Quadratic Restriction Shiing-Shen Chern

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Finsler Geometry Is Just Riemannian Geometry Without the Quadratic Restriction Shiing-Shen Chern Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction Shiing-Shen Chern metric of the domain bounded by an ellipsoid n his Habilitationsvortrag of 1854, Riemann 2 in C , and their calculations showed that the introduced a metric structure in a general Kobayashi metric on such a domain exhibits Ispace based on the element of arc many of the nice properties of a Riemannian met- (1) ds = F(x1,...,xn; dx1,...,dxn) . ric but is plainly not quadratic. There are developments in Finsler geometry Here, F(x, y) is a positive (when y =0) function 6 in recent years which deserve attention. It has on the tangent bundle TM and is homogeneous been shown that modern differential geometry of degree one in y. An important special case is provides the concepts and tools to effect a treat- when ment of Riemannian geometry, without the qua- 2 i j dratic restriction, in a direct and elegant way so (2) F = gij(x)dx dx . that all results, local and global, are included. Historical developments have conferred the This not only gives a better understanding of the name Riemannian geometry to this case while the geometry but opens a vista comparable to the general case, Riemannian geometry without the developments of algebraic geometry from quadratic restriction (2), has been known as quadrics to general algebraic varieties. Finsler geometry. Our report will be divided into the following The name “Finsler geometry” came from sections: Finsler’s thesis of 1918. It is actually the geom- 1. Connections and the equivalence problem etry of a simple integral and is as old as the cal- 2. The second variation of arc length and com- culus of variations. Hilbert attached great im- parison theorems portance to the field, and in his famous Paris 3. Harmonic theory address of 1900 devoted Problem 23 to the vari- 4. Complex Finsler geometry ational calculus of ds and its geometrical over- 5. The Gauss-Bonnet formula tones. A rewrite (usingR Euler’s theorem for ho- These topics do not cover all the important re- mogeneous functions) of the integrand here—the cent developments in Finsler geometry, nor do Hilbert form—will be discussed in the following they highlight the pivotal role played by the section. local theory. For a more comprehensive overview Finsler geometry is not a generalization of of the subject, see the proceedings volume [8] Riemannian geometry. It is better described as (especially the prefaces) and references therein. Riemannian geometry without the quadratic re- striction (2). A special case in point is the inter- Connections and the Equivalence esting paper [11]. They studied the Kobayashi Problem The fundamental problem in local Finsler geom- Shiing-Shen Chern is director emeritus at MSRI, Berke- etry is the equivalence problem: To find a com- ley, California. plete system of invariants or to decide when SEPTEMBER 1996 NOTICES OF THE AMS 959 two Finsler metrics differ by a coordinate trans- been shown that, with respect to the Laplace-Bel- formation. In the Riemannian case this was the trami operator on PTM, the first pair of forms form problem, solved in 1870 by E. B. Christof- has eigenvalue (n 1), the second pair 2(n 2), fel and R. Lipschitz. In his solution Christoffel and so on. Also, the− last item, namely, the −con- introduced the requisite covariant differentia- tact form itself, is harmonic. tion, which Ricci developed into his tensor analy- It turns out that by simply taking the exterior sis, making it a fundamental tool in classical derivative of ω one can define a connection in differential geometry. the pulled-back bundles over PTM. This con- The key idea in Finsler geometry is to consider nection is characterized by being torsion-free, to- the projectivized tangent bundle PTM (i.e., the gether with the property that the scalar product bundle of line elements) of the manifold M. The remains invariant when the canonical section ` main reason is that all geometric quantities con- is parallel displaced. In the Riemannian case structed from F are homogeneous of degree this construct reduces to the Christoffel-Levi- zero in y and thus naturally live on PTM, even Civita connection. For a detailed treatment using though F itself does not. moving frames, see [14]. An explicit formula of To describe one such quantity, let xi, this connection in natural coordinates can be 1 5 i 5 n, be local coordinates on M. Express found in [18] or [10]. i ∂ i i tangent vectors as y = y ∂xi , so that (x ,y ) can The above connection gives connection forms be used as local coordinates of TM and, with yi in the relevant principal bundles and leads to a homogeneous, as local coordinates of PTM. The solution of the equivalence problem in a stan- function F in (1) gives rise to a function dard way. F(x1,...,xn; y1,...,yn), linearly homogeneous Exterior differentiation of the connection in the y’s. The fundamental tensor gij is defined forms gives the notion of curvature, as qua- 1 2 as the y-Hessian ( 2 F )yi yj . Each gij is homo- dratic exterior differential forms. Their compo- geneous of degree zero in y, hence lives on i nents consist of two curvature tensors, Rjkl PTM. In case F2 is quadratic, these g ’s sim- i ij and Pjkl. To get numerical invariants, one con- ply reduce to the usual gij(x)’s of (2), which siders a flag based at x M, with flagpole ∈ then also live on M. 0 = y TxM and a unit length transverse edge The tangent and cotangent bundles, TM and V 6 which∈ is perpendicular to the flagpole. Here, T ∗M respectively, over M, when pulled back to the length of V and its orthogonality to y are i j PTM, have distinguished properties. The object measured by gij(x, y) dx dx . The flag curva- i j ⊗ gij dx dx defines a scalar product on each ture is then K(x, y, V):=Vi(`jR `l)Vk. It ⊗ jikl fibre of these bundles. With respect to this scalar takes the place of the sectional curvature in the product, the pulled-back TM admits a globally Riemannian case. It is also independent of y1 ∂ defined section ` := F ∂xi of unit length. Its nat- whichever of the standard connections (among ural dual is, modulo a rewrite, the integrand of several !) we choose and can even be obtained Hilbert’s invariant integral in the calculus of from a dynamical systems point of view (cf. variations, namely, the one-form Foulon [15]). ∂F (3) ω = dxi . To close this section, let us mention a result ∂yi of Akbar-Zadeh’s [3]: A compact boundaryless We will call it the Hilbert form. This is a global Finsler manifold is locally Minkowskian if and section of the pulled-back T ∗M, and it has unit only if it has zero flag curvature. length. One can also view it as a globally defined linear differential form on PTM. By Euler’s the- The Second Variation of Arc Length and b orem, one can rewrite the length integral a ds Comparison Theorems b on M as a ω. R A systematic study of global Riemannian geom- The HilbertR form is a powerful piece of data. etry began with Heinz Hopf. His 1932 report in Hilbert devoted almost all his discussion of the the Jahresberichte der deutschen Mathematiker 23rd problem to it. As an example of its utility, Vereinigung was a landmark. Hopf and Rinow in- we mention that a typical nondegeneracy hy- troduced the notion of completeness, and in his pothesis (cf. [13, 14]) on F, when expressed in thesis Hopf gave a description of all the Clifford- terms of ω, reads Klein space forms, i.e., the Riemannian manifolds n 1 ω (dω) − =0. of constant sectional curvature. ∧ 6 A natural question in this area is the relation That is, ω defines a contact structure on PTM. between curvature and topology. Of particular As a second example, consider the differen- interest is the study of Riemannian manifolds tial ideal ω, dω , ω dω, (dω)2 , …, whose sectional curvature keeps a constant sign. { } { ∧ } ω (dω)n 1 generated by the Hilbert form. A fundamental tool for this study is a formula { ∧ − } Now, PTM is a Riemannian manifold with a nat- for the second variation of arc length. Here the ural metric induced by the gij’s. In [10] it has remarkable fact is that the same formula holds, 960 NOTICES OF THE AMS VOLUME 43, NUMBER 9 with the sectional curvature in the Riemannian With this Laplacian, which is an elliptic op- case replaced by the flag curvature in the gen- erator, we have Hodge’s theorem: eral Finsler case. As a consequence all the clas- Theorem. On a compact oriented Finsler mani- sical theorems, such as the Hadamard-Cartan fold without boundary, every cohomology class theorem on manifolds of nonpositive curvature, has a unique harmonic representative. The di- the Bonnet-Myers theorem, the Synge theorem, mension of the space of all harmonic forms of de- the first comparison theorem of Rauch as well gree p is the p-th Betti number of the manifold. as the Bishop-Gromov volume comparison the- orem, extend to the Finsler setting. For details, The introduction of the Laplacian opens up a see [5, 6, 13, 17]. whole host of problems, for example Bochner-type There has been a great development on Rie- vanishing theorems and eigenvalue estimates. mannian manifolds of positive curvature. A par- allel theory of Finsler manifolds of positive flag Complex Finsler Geometry curvature deserves to be worked out.
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