P. Gilkey and J. H. Park

Transcription

1 A proof of the Chern-Gauss-Bonnet theorem for indefinite signature metrics using analytic continuation (arxiv: ) P. Gilkey and J. H. Park 22 Aug ICM Satellite Conference on Geometric Analysis

2 Introduction The use of analytic continuation to pass from positive definite metrics to indefinite metrics or to pass between the spacelike and the timelike settings in pseudo-riemannian geometry has proven to be a very fruitful technical tool; results holding in the Riemannian context can often be extended to the pseudo-riemannian context with relatively little additional fuss and results in the spacelike (resp. timelike) settings often give results in the timelike (resp. spacelike) setting. In all cases, one first extends the relevant notions to complex metrics (i.e. sections g to S 2 (T M) R C satisfying det(g) 0) or complex tangent vectors, applies analytic continuation, and thereby passes from one signature to another.

3 Examples Osserman Geometry In the study of Osserman geometry, García-Río et al. [1] examined complex tangent vectors (i.e. elements of TM R C) to show that spacelike Osserman and timelike Osserman were equivalent concepts. [1] E. García-Río, D. N. Kupeli, M. E. Vazquez-Abal, and R. Vazquez-Lorenzo, Affine Osserman connections and their Riemann extensions, Differential Geom. Appl. 11 (1999), E. García-Río M. E. Vazquez-Abal R. Vazquez-Lorenzo

4 Examples Ricci solitons If (M, g, f ) is a gradient Ricci solution and if H f is the Hessian, then one has the identity ( f Ric) + Ric H f = R( f, ) f τ. This was proved in the Riemannian setting; one can use analytic continuation to extend this identity to the indefinite setting

5 Examples Ricci solitons If (M, g, f ) is a gradient Ricci solution and if H f is the Hessian, then one has the identity ( f Ric) + Ric H f = R( f, ) f τ. This was proved in the Riemannian setting; one can use analytic continuation to extend this identity to the indefinite setting The Gauss-Bonnet theorem We shall now derive the Gauss-Bonnet theorem in the indefinite signature (p, q) setting from the corresponding theorem in the definite case; this gives an explanation of the factor of ( 1) p/2 that appears.

6 Gaussian Curvature Intrinsic Geometry Let M be a 2-dimensional Riemannian manifold. If x = (x 1, x 2 ) is a system of local coordinates on M, let g ij := g( xi, xj ). The Gaussian curvature is K := det(g ij ) 1 R 1221 and the volume element is dvol g := det(g ij )dx 1 dx 2. Let Vol(B ɛ (P)) be the volume of the geodesic ball in M of radius ɛ about P M. Then Vol(B ɛ )(P) = πr 2 {1 K(P) 12 r 2 + O(r 4 )}. We refer to Gray [2] for further terms in the expansion. [2] A. Gray, The volume of a small geodesic ball of a Riemannian manifold, Michigan Math. J. 20 (1974), A. Gray

7 Geodesic curvature Let γ be a curve in M and let ν be a unit normal along γ. The geodesic curvature κ g := g( γ γ, ν).

8 Geodesic curvature Let γ be a curve in M and let ν be a unit normal along γ. The geodesic curvature κ g := g( γ γ, ν). Euler characteristic Let χ be the Euler-characteristic of M. If M is triangulated, then χ(m) = #vertices #edges + #triangles +....

9 Geodesic curvature Let γ be a curve in M and let ν be a unit normal along γ. The geodesic curvature κ g := g( γ γ, ν). Euler characteristic Let χ be the Euler-characteristic of M. If M is triangulated, then χ(m) = #vertices #edges + #triangles Extrinsic geometry If T (x 1, x 2 ) is an isometric immersion of M into R 3, then g ij = x i T x j T. If ν := x 1 T x 2 T x1 T x2 is a unit normal, then II ( xi, xj ) := xi xj T ν is the second fundamental form and K = det(ii )/ det(g).

10 The Gauss-Bonnet Theorem Theorem [Gauss-Bonnet (1848)-von Dyck (1888)] Let (M, g) be a 2-dimensional Riemannian manifold with piecewise smooth boundary M. Let κ = R 1221 be the Gaussian curvature, let κ g be the geodesic curvature, and let α i be the interior angles at which consecutive smooth boundary components meet. Then 2π χ(m) = κ + κ g + (π α i ). M M i Gauss Bonnet von Dyck

11 The Chern-Gauss-Bonnet Theorem in general The Euler form (or Pfaffian) is defined by setting: E 2l (g) := R i 1 i 2 j 2 j 1...R i2l 1 i 2l j 2l j 2l 1 g(e i 1... e i 2l, e j 1... e j 2l) (8π) l l! Let ρ kl := g il R ijkl be the Ricci tensor and let τ = g kl ρ kl the scalar curvature. One has: E 2 = τ 4π E 4 = τ 2 4 ρ 2 + R 2 32π 2, E 6 = 1 384π 3 {τ 3 12τ ρ 2 + 3τ R ρ ab ρ ac ρ bc 24ρ ab ρ cd R acbd 24ρ uv R uabc R vabc + 8R abcd R aucv R bvdu 2R abcd R abuv R cduv } Set dvol g = det(g ij ) 1/2 dx 1... dx m..

12 Theorem [Chern [4] see also Allendoerfer and Weil [3]] Let M be a compact Riemannian manifold of dimension m = 2l without boundary. Then M E 2l(g) dvol g = χ(m). [3] C. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Am. Math. Soc. 53 (1943), [4] S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45, (1944) C. Allendoerfer S. Chern A. Weil

13 Extension to the pseudo-riemannian setting Definition (M, g) is a pseudo-riemannian manifold of signature (p, q). Let E 2l (g) := R i 1 i 2 j 2 j 1...R i2l 1 i 2l j 2l j 2l 1 g(e i 1... e i 2l, e j 1... e j 2l) (8π) l. l! Theorem Let (M, g) be a compact pseudo-riemannian manifold of signature (p, q) without boundary. If dim(m) 1 mod 2 or if p 1 mod 2, then χ(m) vanishes. If dim(m) p 0 mod 2, χ(m) = ( 1) p/2 E m dvol g. [5] A. Avez, Formule de Gauss-Bonnet-Chern en métrique de signature quelconque, C. R. Acad. Sci. Paris 255 (1962), [6] S. Chern, Pseudo-Riemannian geometry and the Gauss-Bonnet formula, An. Acad. Brasil. Ci. 35 (1963), (There is a small mistake in the paper of Chern as the factor of ( 1) p is missing). M

14 Examples 1 Let (M, g) be a Riemannian manifold with of dimension m = 2l. Let g = g be a pseudo-riemannian metric of signature (m, 0). Then the Levi-Civita connection is unchanged and the curvature operator is not changed. As R ijkl = g ln R ijk n, we have that R ijkl ( g) = R ijkl (g) and E 2l ( g) = R i1 i 2 j 2 j 1... R i2l 1 i 2l j 2l j 2l 1 g(e i 1... e i l,e j 1... e j l) (8π) l l! = ( 1) l E 2l (g). Since the underlying volume form is not changed, it is necessary to introduce a factor of ( 1) l to correct the Chern-Gauss-Bonnet formula. 2 Let (M i, g i ) be Riemannian manifolds with dim(m i ) = m i = 2l i. Let (M = M 1 M 2, g = g 1 + g 2 ) be a pseudo-riemannian manifold of signature (m 1, m 2 ). As R = R 1 + R 2, E m (g) = E m1 ( g 1 )E m2 (g 2 ) = ( 1) l 1 E m (g 1 + g 2 ).

15 Proof - complex metrics We follow Gilkey-Park [14]. Let g S 2 (T M) C be a complex valued symmetric bilinear form on the tangent space. We assume det(g) 0 as a non-degeneracy condition. The Levi-Civita connection and curvature tensor may then be defined. Set dvol(g) := det(g ij ) 1/2 dx 1 dx m. We do not take the absolute value. There is a subtlety here since, of course, there are 2 branches of the square root function. We shall ignore this for the moment in the interests of simplicity and return to this presently. [14] P. Gilkey and J.H. Park, A proof of the Chern-Gauss-Bonnet theorem for indefinite signature metrics using analytic continuation, arxiv: P. Gilkey J.H. Park

16 Proof - Euler-Lagrange Equations Consider M E m(g) dvol(g). Let h S 2 (T M) C be a perturbation of the metric. We form g ɛ := g + ɛh and differentiate with respect to the parameter ɛ. We then integrate by parts to define the Euler-Lagrange equations: { ɛ E m (g + ɛh) dvol(g + ɛh)} = M M g((te m )(g), h) dvol(g) where TE m (g) S 2 (T M) C is the transgression of the Pfaffian E m. The Chern-Gauss-Bonnet theorem shows that TE m (g) = 0 if dim(m) = m and if g is Riemannian, i.e. if g and h are real since M E m(g + ɛh) dvol(g + ɛh) = χ(m) is independent of h. ɛ=0

17 Proof - Analytic continuation The invariant TE m is locally computable. In any coordinate system, it is polynomial in the derivatives of the metric, the components of the metric, and det(g ij ) 1/2. Thus it is holomorphic in the metric. Thus vanishing identically when g is real implies it vanishes identically when g is complex. Consequently, the Euler-Lagrange equations vanish. In particular, if g ɛ is a smooth 1-parameter family of complex metrics starting with a real metric, then we can define a branch of the square root function along this family so that the following integral is independent of ɛ: E m (g ɛ ) dvol(g ɛ ). M

18 Proof - Decomposing the tangent bundle We may find an orthogonal direct sum decomposition TM = V V + where V is timelike with dim(v ) = p and where V + is spacelike with dim(v + ) = q. This decomposes g = g g + where g is negative definite on V and g + is positive definite on V +. For ɛ [0, π], define: g ɛ = e ɛ 1 g g +. Then g 0 = g g + is a Riemannian metric while g π = g g + is the given pseudo-riemannian metric. We have det(g ɛ ) = e ɛ 1p det( g ) det(g + ) = e ɛ 1p det(g 0 ) so the branch of the square root function along the deformation from g 0 to g π is given by e ɛ 1p/2. Thus if p is odd, this is purely imaginary for g π and hence the Euler characteristic vanishes. If p is even, det(g 0 ) = det(g π ) so ( 1) p/2 det(g 0 ) = ( 1) p/2 det(g). This completes the proof.

19 The Chern-Gauss-Bonnet Theorem for Riemannian manifolds with bd Definition: Let {e 1, e 2,..., e m } be a local orthonormal frame field where e m is the inward unit normal. Let L ab := g( ea e b, e m ) be the second fundamental form. We sum over indices a i and b i ranging from 1 to m 1 to define: { Ra1 a F m,ν := 2 b 2 b 1...R a2ν 1 a 2ν b 2ν b 2ν 1 L a2ν+1 b 2ν+1...L am 1 b m 1 (8π) ν ν! Vol(S m 1 2ν )(m 1 2ν)! g(e a 1 e a m 1, e b 1 e b m 1 ). Theorem Let (M, g) be a compact smooth Riemannian manifold of dimension m. If m is odd, χ(m) = 1 2χ( M). If m is even, χ(m) = E m (g) vol(g) + F m,ν vol(g M ). M ν M [4] S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. 45, (1944) }

20 Examples Let M be a compact smooth Riemannian manifold with boundary of dimension m. 1 If m = 2, then χ(m) = 1 4π M τ + 2 M L aa. 2 If m = 3, then χ(m) = 1 8π M R a 1 a 2 a 2 a 1 + L a1 a 1 L a2 a 2 L a1 a 2 L a1 a 2 = 1 8π M τ M. 3 If m = 4, then χ(m) = 1 32π M (τ 2 4 ρ 2 + R 2 ) π M (3τL 2 aa + 6R amam L bb + 6R acbc L ab +2L aa L bb L cc 6L ab L ab L cc + 4L ab L bc L ac ).

21 Extension to the pseudo-riemannian setting Theorem: Let (M, g) be a compact smooth manifold pseudo-riemannian manifold of even dimension m and signature (p, q) which has smooth non-degenerate boundary M. If m is odd, then χ(m) = 1 2χ( M). If p is odd, then χ(m) = 0. Otherwise { χ(m) = ( 1) p/2 E m (g) vol(g) + } F m,ν vol(g M ). M ν M [7] L. J. Alty, The generalized Gauss-Bonnet-Chern theorem, J. Math. Phys. 36 (1995), doi: / [8] P. Gilkey and J.H. Park, A proof of the Chern-Gauss-Bonnet theorem for indefinite signature metrics using analytic continuation, arxiv:

22 Proof If (M, g) is Riemannian, this follows from the previous result of Chern. So the trick is to extend it to the pseudo-riemannian setting. Again, we will use analytic continuation. But there is an important difference. We fix a non-zero vector field X which points in on the boundary and consider a smooth 1-parameter of complex variations g ɛ so that g ɛ (X, X ) 0 and so that X T ( M) with respect to g ɛ. We may then set ν = X g ɛ (X, X ) 1/2. In the expression for F m,ν, there are an odd number of terms which contain L and hence g ɛ (X, X ) 1/2. Given a pseudo-riemannian metric g, we construct g 0 as above. Let g0,ab M and g M ab be the restriction of the metrics g 0 and g, respectively, to the boundary. If X is space-like, then det(gab M) = ( 1)p det(g0,ab M ) and the analysis proceeds as previously; the whole subtlety arising from the square root of the determinant.

23 Proof continued If X is time-like, then det(gab M) = ( 1)p 1 det(g0,ab M ). However g(x, X ) = g 0 (X, X ) and thus once again, we must take the square root of ( 1) p in the analytic continuation. Apart from this, the remainder of the argument is the same as that used to consider the case of a manifold without boundary.

24 Euler Lagrange Equations Let (M, g) be a Riemannian manifold of dimension m > 2l. We consider E 2l (g) vol(g). M This is no longer a topological invariant but is still of interest. Let h be a symmetric 2-tensor field. One integrates by parts to define the associated Euler-Lagrange equations: { ɛ E 2l (g + ɛh) vol(g + ɛh)} = M M g(te 2l (g), h) vol(g). The transgression TE 2l is canonically defined symmetric 2-tensor. It vanishes identically if 2n = m but in general is non-zero for m > 2n. ɛ=0

25 Euler-Lagrange Equations II A-priori since E 2l involves the 2 nd derivatives of the metric TE 2l could involve one 4 th derivative of the metric or two 3 rd derivatives of the metric. Berger [9] conjectured that this was not in fact the case; that TE 2l only involved the curvature tensor. [9] M. Berger, Quelques formulas de variation pour une structure riemanniene, Ann. Sci. Éc. Norm. Supér. 3 (1970), M. Berger

26 Euler-Lagrange Identities III Berger s conjecture was established by Kuz mina [10] (see related work by Labbi [11], Gilkey, Park, and Sekigawa [12], and A and J Navarro [13]).

27 Euler-Lagrange Identities III Berger s conjecture was established by Kuz mina [10] (see related work by Labbi [11], Gilkey, Park, and Sekigawa [12], and A and J Navarro [13]). Theorem TE 2l = c(m)r i1 i 2 j 2 j 1...R i2l 1 i 2l j 2l j 2l 1 e i 2l+1 e j 2l+1 g(e i 1... e i 2l+1, e j 1... e j 2l+1). [10] G. M. Kuz mina, Some generalizations of the Riemann spaces of Einstein, Math. Notes 16 (1974), [11] M.-L. Labbi, Double forms, curvature structures and the (p, q)-curvatures, Trans. Am. Math. Soc. 357 (2005), [12] P. Gilkey, J.H. Park, and K. Sekigawa, Universal curvature identities, Diff. Geom. Appl., 29 (2011) [13] A. Navarro and J. Navarro, Dimensional curvature identities on pseudo-riemannian geometry, arxiv:

28 Dimension m = 2 One has, for example, that TE 2 = c 2 {R ijji e k e k 2R ijki e j e k } In dimension 2, TE 2 = 0; this yields the familiar identity that ρ = 1 2 τg. Furthermore, τ dvol has a critical point in higher dimensions if and only if ρ = λg so (M, g) is Einstein.

29 Dimension m = 4 If m = 4, then TE 4 = c 4 {(R ijji R kllk 4R ijki R ljkl + R ijkl R ijkl )e n e n +{R klni R klnj 2R knik R lnjl 2R iklj R nkln + R kllk R nijn }e i e j. Since this vanishes in dimension m = 4, this yields the 4-dimensional Eun-Park-Sekigawa identity [15] which has important applications in studying the Euler Lagrange equations for the Chern-Gauss-Bonnet action in gravity. [15] Y. Euh, J. H. Park, K. Sekigawa, A curvature identity on a 4-dim. Riemannian manifold, Results Math. 63 (2013), J.H. Park K. Sekigawa Y. Euh

30 Dimension 6 If m = 6, one obtains the 6-dimensional Euh-Park-Sekigawa identity [17] obtained by setting TE 6 = 0 where TE 6 = ( τ τ R 2 6τ ρ 2 + 8ρ ab ρ bc ρ ca 12ρ ab ρ cd R acbd 12ρ uv R abcu R abcv + 4R abcd R aucv R ubvd 2R abuv R uvcd R abcd )g ij 3τ 2 ρ ij 3 R 2 ρ ij + 12 ρ 2 ρ ij + 12τρ ia ρ ja + 12τρ ab R iabj 6τR iabc R jabc 24ρ ia ρ jb ρ ab 24ρ ac ρ bc R iabj +24ρ aj ρ cd R acid + 24ρ ai ρ cd R acjd + 24ρ ab R icdj R acbd +48ρ cd R iabc R jabd + 6ρ jd R abci R abcd + 6ρ id R abcj R abcd +12R iuvj R abcu R abcv + 12R ibac R jbuv R acuv 24R iabc R jubv R aucv. [17] Y. Euh, J. H. Park and K. Sekigawa, A curvature identity on 6-dimensional Riemannian Manifolds and its applications, (private communication).

31 Euler Lagrange Equations in indefinite signature One can define E 2l and TE 2l in indefinite signature setting: E 2l (g) := R i 1 i 2 j 2 j 1...R i2l 1 i 2l j 2l j 2l 1 g(e i 1... e i 2l, e j 1... e j 2l) (8π) l. l! TE 2l = c(m)r i1 i 2 j 2 j 1...R i2l 1 i 2l j 2l j 2l 1 e i 2l+1 e j 2l+1 g(e i 1... e i 2l+1, e j 1... e j 2l+1). dvol(g) = (det(g ij )) 1/2 dx 1... dx m. { Theorem ɛ E m (g + ɛh) dvol(g + ɛh)} M = g((te m )(g), h) dvol(g). M [18] P. Gilkey, J.H. Park, and K. Sekigawa, Universal curvature identities II, J. Geom and Physics 62 (2012), ɛ=0

32 Proof Rather than proving this result a new as was done in [18], we can derive it from the result in the Riemannian setting using analytic contination. We note that it holds for (g, h) real and then permit (g, h) to be complex. The same argument given previously using the decomposition of TM = V + V then shows any two non-degenerate pseudo-riemannian metrics lie in the same arc component and provides a path between the metric g = g + g and a positive definite metric g + g ; it also gives us a way of analytically continuing the square root function. Appropriate powers of 1 will arise, of course, in comparing dvol with dvol, but these cancel out on both sides of the equation.

33 Generalizations 1. An appropriate extension of the Euler-Lagrange equations in the Riemannian setting to manifolds with boundary (as was done in [19]) then yields the corresponding formulas in the pseudo Riemannian setting with non-degenerate boundary. 2. There are analogous results for the Euler-Lagrange equations in the Kähler setting defined by Chern forms that extend work of Gilkey-Park-Sekigawa [20] for Kähler geometry to pseudo-kähler geometry. [19] P. Gilkey, J.H. Park, and K. Sekigawa, Universal curvature identities III, International Journal of Geometric Methods in Modern Physics 106 (2013), [20] P. Gilkey, J.H. Park, and K. Sekigawa, Universal curvature identities and Euler Lagrange Formulas for Kaehler manifolds, to appear Joint with J. H. Park and K. Sekigawa, to appear J. Math. Soc. of Japan.