Geodesic and curvature of piecewise flat Finsler surfaces

Transcription

1 Geodesic and curvature of piecewise flat Finsler surfaces Ming Xu Capital Normal University (based on a joint work with S. Deng) in Southwest Jiaotong University, Emei, July 2018

2 Outline 1 Background Definition

3 Outline 1 Background Definition 2

4 Outline 1 Background Definition 2 3 Landsberg piecewise flat Finsler surface and a combinatoric Gauss-Bonnet formula

5 Outline 1 Background Definition 2 3 Landsberg piecewise flat Finsler surface and a combinatoric Gauss-Bonnet formula 4

6 Outline 1 Background Definition 2 3 Landsberg piecewise flat Finsler surface and a combinatoric Gauss-Bonnet formula 4 5

7 Ming XuCapital - Meanwhile, Normal University(based thereon are a joint important work with S. Deng)Geodesic applications and curvature inofcomputer piecewise flat Finsler surfaces Background Definition The study on piecewise flat Riemannian geometry has a long history: - It deals with familiar geometrical objects like polytopes, and is deeply related to the study of topology. - In 1961, T. Regge introduced the notion of piecewise flat Riemannian and pseudo-riemannian manifolds, and proposed studying its geometry. - In 1984, J. Cheeger, W. Müller and R. Schrader discussed Regge s calculus (using piecewise flat geometry to approximate smooth geometry) and the combinatoric presentations for topological invariants. - In recent years, many analytic theories of Riemannian geometry are generalized to the piecewise flat context.

8 Background Definition However, piecewise flat Finsler geometry has not been touched. So we can do a lot of things on this new field: - Set up the theoretical framework, generalized from smooth Finsler geometry. - A lot of new problems and theories to be explored. - Suching for its applications.

9 Background Definition Before defining piecewise flat Finsler manifold, we need a very brief warmup for Finsler geometry. Definition A Minkowski norm on a real vector space V is a continuous function F : V [0, + ) which is positive, smooth, and positively homogeneous of degree one on V\{0}, and most importantly, the Hessian matrix (g ij (y)) = 1 2 [F 2 ] y i y j with respect to any linear coordinates y = y i e i is positive definite whenever y is not zero.

10 Background Definition Definition Given a smooth manifold M, we provide each tangent space T x M with a Minkowski norm F(x, ) which depends smoothly on x M, then it is a Finsler manifold or Finsler space. The function F : TM [0, + ) is then called a Finsler metric.

11 Background Definition For a Minkowski norm, or a Finsler metric, the Hessian (g ij ) = ( 1 2 [F 2 ] y i y j ) defines a inner product which depends on the choice of the nonzero vector y, i.e. 2 u, v F y = 1 2 t s F 2 (y + tu + sv) t=s=0 = g ij (x, y)u i v j, for y = y i x i 0, u = u i x i and v = v i y i.

12 Background Definition Given a Finsler manifold (M, F). For each x M, the inner products, F y defines a Riemannian metric on T x M\{0}.

13 Background Definition In T x M, there is a special hypersurface {y T x M with F(y) = 1}, called the indicatrix. The Riemannian metric, F y tells us how to calculate the volume/length on the indicatrix, which provides the notion of angle at x M.

14 Background Definition In this talk, we will mainly deal with a special class of flat Finsler manifold, called the Minkowski space, i.e. a region in a real vector space, endowed with a Finsler metric which is invariant under parallel shifting. It is stand and flat in the sense that its flag curvature is identically zero, geodesics are straight lines/rays/segments, and more generally all totally geodesic submanifolds are "flat things".

15 Background Definition Now we can define piecewise flat Finsler manifold. Definition Let M be a triangulated manifold, with all the k-dimensional simplices denoted as {S k,α, α A k }, for each k from 0 to n = dim M. For each S k,α, we choose a Minkowski norm F k,α such that (S k,α, F k,α ) is Minkowski space. Whenever we have S k,α S k,β, then we require F k,α = F k,β Sk,α. Then we call (M, {S k,α }, {F k,α }) a piecewise flat Finsler space.

16 Background Definition The fundamental model: the boundary surface of a polytope in a 3-dimensional Euclidean space, which is a piecewise flat Riemannian surface, or the boundary surface of a polytope in a 3-dimensional Minkowski space, which is a piecewise flat Finsler surface.

17 Example: a polytope in (R^3,F)

18 Background Definition In the following discussion, we further assume M is closed (i.e. compact and no boundary), and we only consider the case that dim M = 2, i.e. M is a piecewise flat surface. For a piecewise flat Finsler surface M, there are only three classes of simplices: (2-dimensional) triangles, (1-dimensional) edges and (0-dimensional) vertices. To define the metric, we only need to point out the Minkowski norm on each triangle, and require them to coincide if two triangles have a common edges.

19 Background Definition Notice that the only singularities of a piecewise flat Riemannian surface are vertices. But for a piecewise flat Finsler surface, the edges may also be singular, because the Minkowski norms for its two sides may not be the same. So in this sense, piecewise flat Riemannian geometry and piecewise flat Finsler geometry are very different.

20 Background Definition The "tangent space" T x M can be defined for any x M, which is an infinitely enlargement of a local neighborhood of x. There are three possibilities: - When x is inside a triangle (i.e. not on any edge, not a vertex), then T x M is a Minkowski plane. - When x is inside an edge (i.e. not a vertex), then T x M consists of two Minkowski half-plane with a common boundary straight line. - When x is a vertex, then T x M consists of finitely many Minkowski cones with a common vertex, {C x,1,..., C x,m }, C x,i C x,i+1 is the ray R x,i (R x,0 = R x,m ). In all above cases, T x M is called a tangent cone in general.

21 The vertex x in a piecewise flat Finsler surface M Locally isometric The tangent cone T_xM (C_2,F_2) (C_1,F_1) x x (C_3,F_3) (C_4,F_4)

22 Background Definition To distinguish it with the tangent space in the smooth geometry, we call T x M a tangent cone. The tangent vector is a arrow lay on a tangent cone, from or to the vertex x. So there are two types of tangent vectors at each x, incoming or outgoing.

23 Let M be a piecewise flat Finsler surface. The metrics on the triangles patch up to define a non-reversible and non-smooth metric on the manifold. Applying the local minimizing principle for the arc length functional defined for all piecewise linear paths, the geodesic can be defined. To study the local behavior of geodesics, we only need to look at the tangent cone.

24 When x is inside a triangle, because the metric is the flat Minkowski metric, the geodesics passing x are simple: straight lines. When x is inside an edge, the geodesics are determined by the Snell-Descartes law, which can be formulated as the following theorem:

25 Theorem Let (T i, F i ) with i = 1, 2 be two Minkowski half-plane with the common line E, such that F 1 E = F 2 E. Then we have the following: (1) A unit speed curve R x,u1 R x,u2 where x E, u 1 is incoming (outgoing) and u 2 is outgoing (incoming) is a geodesic iff u 1, v F 1 u 1 = u 2, v F 2 u 2, where the nonzero vector v is tangent to E. (2) Any Ray R x,u1 T 1 with x E can be uniquely extended to the other side, to be a unit speed geodesic. (3) For any two points x i T i, there exists a unique unit speed geodesic from x 1 to x 2.

26 The edge crossing equation: <u1,v>_{u1}=<u2,v>_{u2} (C1,F1) u1 E v (C2,F2) u2

27 The method for proving this theorem is variational method, like discussing refraction in optics. But here is no total reflection.

28 When x is vertex, it is the most interesting. A geodesic reaching a vertex may not be able to be extended, or it may have infinitely many extension (which are still geodesics).

29 Now we consider the case that x is a vertex. We can prove there are only three types of geodesics in a tangent cone T x M when x is a vertex. Theorem A maximally extended geodesic in a tangent cone T x M must be one of the following: (1) One ray R x,u, passing x, in which u is either incoming or outgoing. (2) Two rays R x,u1 R x,u2, passing x, in which u 1 is incoming and u 2 is outgoing. (3) The union of finitely many line segments and two rays, not passing x. In particular, a geodesic not passing x in T x M can not cross infinitely many edges of T x M.

30 Case (1): one ray passing the vertex, in or out Case (3): Two rays and finite segments, not passing the vertex Case (2): two rays passing the vertex, one ray in and one ray out Impossible case: infinite crossings with the edges.

31 Observation from this theorem: - We want to distinguish the geodesics in (1) and (2), i.e. explicitly show when a ray can be extended as a geodesic at a vertex. - For any ray c passing x, we can perturb it leftward or rightward by paralell shifting, which provides c + and c of type (3), i.e. each of c ± consists of a finite union of segments and two rays. We call the newly emerged ray (not the one by perturbing c) the other end of c ±.

32 C+ c c- Two perturbations for the ray c passing the vertex in the tangent cone

33 Ignoring all the technical details and notations, we proved this. Theorem (1) The ray can not be extended as a geodesic iff c ± intersect. (2) The ray can be uniquely extended as a geodesic iff c ± does not intersect and the other ends of c ± are parallel. (3) The ray have infinitely many ways to be extended as a geodesic iff c ± does not intersect, and the other ends of c ± are not parallel.

34 C+ The other end of c- c c- The other end of c+

35 c+ and c- do not intersection, and the other ends of them are not parallel, iff infinite many extensions of the ray c as geodesics passing the vertex in the tangent cone, iff K<0 where K is the measure of all possible extensions. C+ c c- All these are possible extensions of c as geodesics

36 C+ c The unique extension of c as geodesic c- c+ and c- do not intersection, and the other ends of them are parallel, iff there is an unique extension of the ray c as geodesic passing the vertex in the tangent cone, iff K=0 where measure is zero implies the uniqueness.

37 C+ c c- c+ and c- intersects iff there is no extension of c as a geodesic iff K>0 where K is the measure for the angle corresponding to the dotted rays, but they are not extensions of c as geodesics, and K is the virtual measure of all the extensions.

38 Landsberg piecewise flat Finsler surface and a combinatoric Gauss At any x M, angles at x can be defined using the measure (induced by the Hessian, or the fundamental tensor) on the indicatrices S ± x M in the tangent cone T x M, where ± marks outgoing and incoming. Using this notion of angle at x M, we can provide the following conceptional definition for curvature. Notice that the precise definition requires many technical details, which I have to skip.

39 Landsberg piecewise flat Finsler surface and a combinatoric Gauss Definition Let c ± be the two perturbation for the ray R x,u, where x is a vertex, and u is a outgoing or incoming unit tangent vector at x. Then the curvature K (x, u) is the algebraic counting for the angle between the other ends of c ±. Equivalently speaking, K (x, u) is the measure for all the possibilities of extending the Ray R x,u as a unit speed geodesic.

40 Landsberg piecewise flat Finsler surface and a combinatoric Gauss When K (x, u) > 0, the (virtual) measure of all the extensions for R x,u is negative, so R x,u can not be extended in this case (corresponding to (3) of the previous theorem). When K (x, u) = 0, the measure is 0, but it does not correspond to "the empty set", but "one single point on the indicatrix", i.e. "the unique single direction for extending a ray".

41 c+ and c- do not intersection, and the other ends of them are not parallel, iff infinite many extensions of the ray c as geodesics passing the vertex in the tangent cone, iff K<0 where K is the measure of all possible extensions. C+ c c- All these are possible extensions of c as geodesics

42 C+ c The unique extension of c as geodesic c- c+ and c- do not intersection, and the other ends of them are parallel, iff there is an unique extension of the ray c as geodesic passing the vertex in the tangent cone, iff K=0 where measure is zero implies the uniqueness.

43 C+ c c- c+ and c- intersects iff there is no extension of c as a geodesic iff K>0 where K is the measure for the angle corresponding to the dotted rays, but they are not extensions of c as geodesics, and K is the virtual measure of all the extensions.

44 Landsberg piecewise flat Finsler surface and a combinatoric Gauss Following this idea, curvature can also be given at non-vertex points. But it is straight forward that K (x, u) = 0 when x is not a vertex. So a subdivision of M with Minkowski norms induced by the old ones do not change the geodesics as well as the curvature. This is consistent with the obvious fact that subdivision does change the metric of a piecewise flat Finsler manifold).

45 Landsberg piecewise flat Finsler surface and a combinatoric Gauss The curvature defined here is of Riemannian type, like the flag curvature in Finsler geometry, because K (x, u) depends on the vector u. Notice u can be incoming or outgoing, there are two types of curvature in the piecewise flat context. More precisely, K (x, u) is the generalization of the curvature form, as you may see from the Landsberg case.

46 Landsberg piecewise flat Finsler surface and a combinatoric Gauss Definition We call a piecewise flat Finsler surface M to be Landsberg if for any two triangles (T 1, F 1 ) and (T 2, F 2 ) with a common edge E, the crossing-edge equation u 1, v F 1 u 1 = u 2, v F 2 u 2, for any tangent vector v of E, the correspondence between u 1 and u 2 is an isometry between two indicatrices. Notice that the metric on indicatrices are defined by the Hessians (fundamental tensors).

47 Landsberg piecewise flat Finsler surface and a combinatoric Gauss Definition We call a piecewise flat Finsler surface M Berwald if it is Landsberg and the correspondence between u 1 and u 2 in the previous definition is induced by a real linear isomorphism.

48 Landsberg piecewise flat Finsler surface and a combinatoric Gauss Notice that in Finsler geometry, there is a nonlinear parallel moving for tangent vectors. The Landsberg condition can be conceptional described by the property that the nonlinear parallel moving is an isometry between indicatrices. The nonlinear parallel moving is linear when the metric is Berwald. So the above definitions are natural definitions in Finsler geometry.

49 Landsberg piecewise flat Finsler surface and a combinatoric Gauss Theorem (Curvature and Guass-Bonnet for Landsberg piecewise surfaces) Let M be a connected piecewise flat Finsler surface of Landsberg type. Then we have the following: (1) The length of the indicatrix for the Minkowski norm of each triangle is a constant θ independent of the triangle. (2) The curvature K (x, u) only depends on the vertex x, not on the choice of the tangent vector u. In particular the curvature for incoming vectors is the same as that for outgoing vectors (so we can denote it as K (x) for simplicity). (3) The lengths l ± (x) of the indicatrix at any vertex for incoming tangent vectors and that for outgoing tangent vectors are the same. (4) Curvature formula: K (x) = θ l ± (x). (5) Combinatoric Gauss-Bonnet: K (x) = θ χ(m).

50 - Define the Riemannian curvature for piecewise flat Finsler manifolds of high dimensions. - Define the non-riemannian curvature for piecewise flat Finsler manifolds of all dimensions. - Define Landsberg and Berwald type for high dimensions, and study their geometrical and curvature properties. - Find the combinatoric term generalizing the curvature form of a smooth Landsberg space, and prove a combinatoric Gauss-Bonnet-Chern formula.

51 Reference: [1] M. Xu and S. Deng, Geodesic and curvature of piecewise flat Finsler surfaces, Journal of Geometric Analysis, Journal of Geometric Analysis, vol. 28, issue 2 (2018),

52 This is the end of my talk. Sincerely thank organizer of this conference for inviting me to give this talk, and thank the audience for your patience. I would like to dedicate this talk to my friends, Huibin Chang, Ju Tan and Lei Zhang. The communication with them inspired this work. Also sincerely thank the referee for many precious advices, thank Fuquan Fang for helpful discussions and suggestions, thank Joseph A. Wolf and Wolfgang Ziller for useful communication.