Finsler Geometry Is Just without the Quadratic Restriction Shiing-Shen Chern

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 TM and is homogeneous been shown that modern 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 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 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 forms, i.e., the Riemannian 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 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. As a start, It is possible that Finsler geometry will be most the case of constant positive flag curvature is useful in the complex domain, because every gradually being understood; see Shen [18]. complex manifold, with or without boundary, has Since PTM plays a central role in Finsler a Caratheodory pseudo-metric and a Kobayashi geometry, the most fundamental invariant pseudo-metric. Under favorable (though some- 2 should be the Ricci scalar Ric. It is a scalar func- what stringent) conditions these are C metrics ik j l and, most importantly, they are naturally Fins- tion on PTM and is defined as g (` Rjikl ` ). Its companion (cf. [4]) is the Ricci tensor lerian. The analysis on the manifold is thus in- 1 2 timately tied to the geometry. For an explicit ex- Ric := ( F Ric) j k. jk 2 y y ample of the Kobayashi metric, see [11]. A natural question is: Can every manifold be Complex Finsler geometry is extremely beau- given a Finsler metric with constant Ricci scalar tiful. Again the bundle of line elements PTM or perhaps one whose Ricci scalar does not de- plays the important role. The scalar product on pend on y? It is well known that, in dimension the pulled-back TM gives rise to a hermitian 3, such a Riemannian metric does not always structure on the complexification of the latter. exist.≥ For some interesting (albeit preliminary) Here the geometrical properties mix well with the considerations in the Finsler case, see the arti- complex structure; connection forms are of type cle [4] by Akbar-Zadeh. (1,0) and curvature forms are of type (1,1). A real- The study of the deformation theory of Finsler valued holomorphic curvature, as a function on structures is an equally worthwhile endeavor PTM, can be introduced. and should also be pursued. From this viewpoint an important class of complex manifolds consists of those whose Harmonic Theory Kobayashi metric has constant holomorphic cur- A fundamental work in global Riemannian geom- vature. When it is a negative constant or zero, etry is Hodge’s theory of harmonic forms. Briefly they have been studied by Abate and Patrizio, it says that on a compact Riemannian manifold cf. [2]. The case of positive constant holomorphic without boundary every cohomology class has curvature deserves investigation. a unique harmonic representative. The main in- gredient is the definition of the Laplace opera- The Gauss-Bonnet Formula tor (on M, not on PTM), since a form is har- Among the relationships between curvature and monic when it is annihilated by the Laplacian. topology is the Gauss-Bonnet formula. In his D.∆ Bao and B. Lackey succeeded in formulat- thesis published in 1925 Hopf proved this for- ing the definition of a Laplacian for Finsler man- mula for an orientable closed hypersurface of ifolds which reduces to the standard one in the even dimension in a Euclidean space, express- Riemannian case. Their work makes use of an un- ing its Euler characteristic as an integral of a cur- usual scalar product on the p-forms on the man- vature function. As a result of using tubes, ifold M. If d∗ denotes the adjoint of the exte- C. B. Allendoerfer and W. Fenchel extended it to rior differentiation operator relative to this scalar an arbitrary submanifold in Euclidean space product, then the Laplacian is defined by the (1940). In 1943 Allendoerfer and Weil extended usual formula the formula to a Riemannian polyhedron; their study of the boundary solid angles is masterful. = d d∗ + d∗ d, ◦ ◦ It turns out that the key idea lies in the con- sideration of the bundle SM of the unit tangent where d is ordinary∆ exterior differentiation. Their definition of the scalar product comes vectors of M, as Chern showed in his proof of from a careful study of the Hodge star operator the Gauss-Bonnet formula on PTM and involves integration over the fibers = χ(M) of PTM. For details cf. [9]. ZM − Ω

SEPTEMBER 1996 NOTICES OF THE AMS 961 by transgression. The extension of the Gauss- In addition to what is yet to be done on the Bonnet formula to Finsler geometry has been subject in order for some obvious questions to considered in [7], in which the relevant curvature be answered, I am inclined to think that future forms were introduced and studied. Several fac- developments lie in further generalizations. The tors occur in this generalization. The most no- geometry of a metric space is always an attrac- table one is the role played by the total volume tive subject. Finsler geometry has been studied of SxM at each point x M. A Gauss-Bonnet for- from this vantage point by A. D. Alexandrov [1], mula is obtained whenever∈ this volume function H. Busemann [12], and M. Gromov [16]. A com- is constant (on M), though its value is not nec- bination of geometric and analytic methods re- essarily equal to that of the Riemannian case. mains a challenging open field. Thus Finsler geometry has features to make the This brief report has addressed the raison study of this problem interesting. d’être of Finsler metrics. For example, The relation of other characteristic classes, a. In the function theory of several complex particularly the Pontryagin classes, to curvature variables, the Kobayashi and Caratheodory remains to be studied. metrics are naturally Finslerian and user friendly; they also render holomorphic map- Conclusion pings distance decreasing. I believe I have shown that almost all the results b. For comparison theorems except Topono- of Riemannian geometry can be developed in the gov’s, the effortless replacement of sectional Finsler setting. It is remarkable that one needs curvature by flag curvature extends their only some conceptual adjustment, no essential validity to the Finsler category. Admittedly, new ideas being necessary. This not only im- despite Akbar-Zadeh’s seminal results [3] plies more general results but also gives a bet- the concept of space forms is more com- ter geometrical understanding. plicated than that in Riemannian geometry, At this point it might be interesting to quote but workers in the field seem to embrace this Riemann: as a refreshing challenge. In space, if one expresses the location c. Recent work on Hodge theory has brought of a point by rectilinear coordinates, forth some natural but unorthodox elliptic operators, together with some clues as to i 2 then ds = (dx ) . Space is there- which geometrical data are more important fore includedq in this simplest case. than others. The next simplestΣ case would per- Finslerian constructs also assert themselves haps include the manifolds in which in applications, most notably in control theory, the line element can be expressed as mathematical biology/ecology, and optics. Nev- the fourth root of a differential ex- ertheless, in spite of the above arguments about pression of the fourth degree. Inves- the relevance and timeliness of the Finslerian tigation of this more general class viewpoint, Riemannian geometry will remain a would actually require no essential most important chapter of Finsler geometry. different principles, but it would be Finally, I would like to thank Steve Krantz rather time-consuming [zeitraubend, and Hugo Rossi for their suggestions. in the original German] and throw relatively little new light on the study References of Space, especially since the results Finsler geometry has an enormous literature. cannot be expressed geometrically. These references include only papers referred to in this article. As is well known, Riemannian geometry can [1] A. D. Alexandrov, A theorem on triangles in a be handled, elegantly and efficiently, by tensor metric space and some applications, Trudy Math. analysis on M. Its handicap with Finsler geom- Inst. Steklov 38 (1951), 5–23 (in Russian). etry arises from the fact that the latter needs [2] M. Abate and G. Patrizio, Finsler metrics—A more than one space, for instance PTM in ad- global approach, Lecture Notes in Math., vol. 1591, dition to M, on which tensor analysis does not Springer-Verlag, 1994. fit well. However, this problem can be remedied [3] H. Akbar-Zadeh, Sur les espaces de Finsler à cour- by working on TM and making sure that all bures sectionnelles constantes, Acad. Roy. Belg. constructions are invariant under rescaling in y. Bull. Cl. Sci. (5) 74 (1988), 281–322. Riemann’s emphasis on Riemannian geome- [4] ———, Generalized Einstein manifolds, J. Geom. Phys. 17 (1995), 342–380. try could be based on the Pythagorian nature of [5] L. Auslander, On curvature in Finsler geometry, the metric. His allusion to general Finsler geom- Trans. Amer. Math. Soc. 79 (1955), 378–388. etry was a remarkable insight. After more than [6] D. Bao and S.S. Chern, On a notable connection a century of mathematical development, his vi- in Finsler geometry, Houston J. Math. 19 (1) (1993), sion was justified. 135–180.

962 NOTICES OF THE AMS VOLUME 43, NUMBER 9 [7] ———, A note on the Gauss-Bonnet theorem for Finsler spaces, Ann. Math. 143 (1996), 233–252. [8] D. Bao, S.S. Chern, and Z. Shen, editors, Finsler geometry (Proceedings of the Joint Summer Re- search Conference on Finsler Geometry, July 16–20, 1995, Seattle, Washington), Cont. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996. [9] D. Bao and B. Lackey, A Hodge decomposition the- orem for Finsler spaces, C. R. Acad. Sci. Paris, to appear. [10] ———, Special eigenforms on the sphere bundle of a Finsler manifold, Cont. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 67–78. [11] B. Blank, D. Fan, D. Klein, S. Krantz, D. Ma, and M.-Y. Pang, The Kobayashi metric of a complex el- 2 lipsoid in C , Experimental Math. 1 (1992), 47–55. [12] H. Busemann, On curvature in two di- mensional Finsler spaces, Ann. Mat. Pura Appl. 31 (1950), 281–295. [13] S.S. Chern, On Finsler geometry, C. R. Acad. Sci. Paris 314 (1992), 757–761. [14] ———, Riemannian geometry as a special case of Finsler geometry, Cont. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 51–58. [15] P. Foulon, Géométrie des équations différentielles du second ordre, Ann. Inst. Henri Poincaré 45 (1) (1986), 1–28. [16] M. Gromov, Sign and geometric meaning of cur- vature, Rend. Sem. Mat. Fis. Milano 61 (1991), 9–123 (1994). [17] Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math., to ap- pear. [18] ———, Finsler manifolds of constant positive cur- vature, Cont. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 83–93.

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