Basic Riemannian Geometry - Department of Mathematical ...

More documents

Recommendations

Info

4.2.5 Baking the cakeEquation (4.4) readsln(a ξ /S n−1κ ) ′ ≤ 0so that, a ξ /S n−1κis decreasing and, in view of the initial conditions,a ξ ≤ S n−1κ .Thus:V (m, r) =≤This is Bishop’s theorem.∫S n−1 ∫ min(c(ξ),r)0∫ min(c(ξ),r)∫S n−10a ξ drdξSκn−1 drdξ = V κ (r).Our final statement is due to Gromov [2] and is a consequence of a simplelemma:Lemma ([2]). If f, g > 0 with f/g decreasing thenis decreasing also.∫ r0f/∫ r0gWith this in hand, we see that, for r 1 < r 2 ,∫ r10a ξ dr/∫ r10Sκn−1 dr ≤∫ r20a ξ dr/∫ r20Sκn−1 dr.Integrating this over S n−1 , noting that the denominators are independentof ξ, gives finally that V (m, r)/V κ (r) is decreasing.References[1] I. Chavel, Riemannian geometry—a modern introduction, CambridgeUniversity Press, Cambridge, 1993. 1[2] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernelestimates for functions of the Laplace operator, and the geometry ofcomplete Riemannian manifolds, J. Differential Geom. 17 (1982), no. 1,15–53. 28, 2828
[3] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Springer-Verlag, Berlin, 1987. 1, 1, 24[4] J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140(1994), no. 3, 655–683. 22[5] M. Spivak, Calculus on manifolds. A modern approach to classical theoremsof advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam,1965. 1[6] F.W. Warner, Foundations of differentiable manifolds and Lie groups,Scott, Foresman and Co., Glenview, Ill.-London, 1971. 1, 2, 1129
Page 3 and 4: 4. An open subset of a manifold is
Page 5 and 6: We note:1. Each df m is a linear ma
Page 7 and 8: 1.4 ConnectionsWe would like to dif
Page 9 and 10: 2. Let S n ⊂ R n+1 be the unit sp
Page 11 and 12: In particular, we conclude that∆f
Page 13 and 14: 2. a Riemannian measure dA;3. a uni
Page 15 and 16: Proposition. For c : [a, b] → M a
Page 17 and 18: 3.3.2 The Gauss LemmaLet ξ, η ∈
Page 19 and 20: for all ξ, η ∈ M m , m ∈ M. S
Page 21 and 22: whenceTake an inner product with U
Page 23 and 24: Theorem. Any manifold of dimension
Page 25 and 26: we have:∫∫∆r dV =Ω t,ɛ[t,t+
Page 27: Two applications of the Cauchy-Schw

Basic Riemannian Geometry - Department of Mathematical ...

Create successful ePaper yourself

Delete template?

Save as template?