Basic Riemannian Geometry - Department of Mathematical ...

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(Recall that ∂ 1|m , . . . , ∂ n|m form a basis for M m .)Now let (g ij ) be the matrix inverse to (g ij ). Then the formula (2.1) for ∇reads:∑Γ k ij = 1 2g kl (∂ i g jl + ∂ j g li − ∂ l g ij ) (2.2)since the bracket terms [∂ i , ∂ j ] vanish (exercise!).l2.2 Differential operatorsThe metric and Levi–Civita connection of a Riemannian manifold are preciselythe ingredients one needs to generalise the familiar operators of vectorcalculus:The gradient of f ∈ C ∞ (M) is the vector field grad f such that, for Y ∈Γ(T M),g(grad f, Y ) = Y f.Similarly, the divergence of X ∈ Γ(T M) is the function div f ∈ C ∞ (M)defined by:(div f)(m) = trace(ξ → ∇ ξ X)Finally, we put these together to introduce the hero of this volume: theLaplacian of f ∈ C ∞ (M) is the function∆f = div grad f.In a chart (U, x), set g = det(g ij ). Thengrad f = ∑ i,jg ij (∂ i f)∂ jand, for X = ∑ i X i∂ i ,div X = ∑ i(∂i X i + ∑ jΓ i ijX j)= √ 1 ∑∂ j ( √ gX j ).gjHere we have used ∑ i Γi ij = (∂ j√ g)/√ g which the Reader is invited todeduce from (2.2) together with the well-known formula for a matrix-valuedfunction A:d ln det A = trace A −1 dA.10
In particular, we conclude that∆f = √ 1 ∑∂ i ( √ gg ij ∂ j f) = ∑ gi,ji,jg ij (∂ i ∂ j f − Γ k ij∂ k f).2.3 Integration on Riemannian manifolds2.3.1 Riemannian measure(M, g) has a canonical measure dV on its Borel sets which we define in steps:First let (U, x) be a chart and f : U → R a measureable function. We set∫ ∫f dV = (f ◦ x −1 ) √ g ◦ x −1 dx 1 . . . dx n .Ux(U)Fact. The change of variables formula ensures that this integral is welldefinedon the intersection of any two charts.To get a globally defined measure, we patch things together with a partitionof unity: since M is second countable and locally compact, it follows thatevery open cover of M has a locally finite refinement. A partition of unityfor a locally finite open cover {U α } is a family of functions φ α ∈ C ∞ (M)such that1. supp(φ α ) ⊂ U α ;2. ∑ α φ α = 1.Theorem. [6, Theorem 1.11] Any locally finite cover has a partition ofunity.Armed with this, we choose a locally finite cover of M by charts {(U α , x α )},a partition of unity {φ α } for {U α } and, for measurable f : M → R, set∫f dV = ∑ ∫φ α f dV.Mα U αFact. This definition is independent of all choices.11
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: 2. Let S n ⊂ R n+1 be the unit sp
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 and 28: Two applications of the Cauchy-Schw
Page 29: [3] S. Gallot, D. Hulin, and J. Laf

Basic Riemannian Geometry - Department of Mathematical ...

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