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My Curvature Conventions My Curvature Conventions Hangjun Xu September 2011 1 Connection on a Vector Bundle Let M be a smooth manifold, and let E be an Rn-vector bundle over M, with a connection D. Let µ1; ··· ; µn be local frame over an open subset U ⊂ M, and regard (µ1; ··· ; µn) as row vector at each point of U, we have the Christoffel symbols of the connection D defined as: k ∗ D @ µj =: Γijµk: @xi P j 1 n t For a section s, it can be written locally as sjU = j a µj = (µ1; ··· ; µn)·(a ; ··· ; a ) .Thus j j j D(a µj) = (da )µj + a Aµj; k k i where A is a connection matrix with entries 1-forms, defined as Aj := Γijdx , and k Aµj = µkAj . Therefore k i k Dµj = Γijdx ⊗ µk = µkAj : k ∗ row Note that each Aj 2 T MjU , with Acolumn. We can then write a connection D as P j D = d + A. For sjU = j a µj, we have j 1 n t 1 n t D(a µj) = (µ1; ··· ; µn) · (da ; ··· ; da ) + A · (a ; ··· ; a ) ; i.e., A acts on the left on the coefficient column vector (a1; ··· ; an). 2 Connection on Endomorphism Bundle With this notation, we can verify that using the induced connection De on End(E) ∼= E ⊗ E∗ is given by De = d + [A; ·], where A is the connection matrix of D. Let ∗ k Note that Chen defines connection matrix first, then defines Γij, but I will do the converse. 1 i ∗ σ := σjµi ⊗ µj be a local section of End(E), where fµj g is the dual frame. We compute: i ∗ i ∗ i k ∗ i ∗ k ∗ D(σjµi ⊗ µj ) = (dσj)µi ⊗ µj + σjAi µk ⊗ µj + σjµi ⊗ ((A )j µj ); (1) where A∗ is a connection matrix of the induced connection on E∗. Note that ∗ ∗ ∗ ∗ j ∗ i 0 = d(µi; µj ) = (Dµi; µj ) + (µi;D µj ) = Ai + (A )j; hence A∗ = −At. Therefore (1) gives: i ∗ i k i i k ∗ D(σjµi ⊗ µj ) = dσj + σj Ak − σkAj µi ⊗ µj i i i ∗ = dσj + (A · σ)j − (σ · A)j µi ⊗ µj = dσ + [A; σ]: 3 Curvature of Vector Bundle We define the curvature of a connection D to be F := D ◦ D : Ω(E) −! Ω2(E). For µ 2 Γ(E), we have F (µ) = (d + A) ◦ (d + A)(µ) = (d + A)(dµ + A · µ) = (dA) · µ + (A ^ A) · µ. j More explicitly, suppose (µ1; ··· ; µn) is a local frame, and µ := a µj as above, we have: j j j k F (a µj) = (d + A) (da )µj + Aka µj j k j j k l j k = (d ◦ da )µj + Aj (da )µk + d(Aka )µj + Aj ^ Aka µl k j j k j k l j k = Aj (da )µk + (dAk)a µj − Ak ^ (da )µj + Aj ^ Aka µl j k j k = (dAk)a + (A ^ A)ka µj: Thus F = dA + A ^ A, acting on the coefficient column from the left, and on the i k k local frame on the right. If we write A = Aidx , i.e., (Ai)j = Γij, then i i j F = d(Aidx ) + (Aidx ) ^ (Ajdx ) @A = i dxj ^ dxi + A · A dx ^ dxj @xj i j i 1 @A @A = j − j + [A ;A ] dxi ^ dxj: 2 @xi @xi i j 2 Notice that F is defined locally, but it turns out to be a global object. F can be regarded as a section of Ω2(End(E)), that is F is a endomorphism bundle-valued 2- form, or from another perspective, a matrix with entries 2-forms. From section 2, we have an induced connection De on the endomorphism bundle, hence we can compute DFe as follows: DFe = dF + [A; F ] = d(A ^ A) + [A; dA + A ^ A] = 0: That is, DFe = 0, i.e., dF = [F; A]. This is called the (second) Bianchi identity. We now want to express F in terms of the Christoffel symbols. We first define the curvature operator R considered as an element of Ω2(End(E)) as follows: F : Ω(E) −! Ω2(E); µ 7! R(·; ·)µ, k k and we define R(@i;@j)µl =: Rlijµk. Thus Rlij acts on µk from the right. We have: ! 1 @Γk Γk R(·; ·)µ = F µ = jl − il + Γk Γm − Γk Γm dxi ^ dxj ⊗ µ : l l 2 @xi @xj im jl jm il k Therefore @Γk Γk Rk = jl − il + Γk Γm − Γk Γm: lij @xi @xj im jl jm il An important property that the curvature operator R satisfies is that R(X; Y )µ = [DX ;DY ]µ − D[X;Y ]µ, for all vector fields X; Y , and all section µ 2 Γ(E). 4 Tangent Bundle Case We now restrict our discussion to (M; g) a Riemannian manifold, and E =TM is the @ tangent bundle, and D is the Levi-Civita connection r on TM. Taking f @xi g to be our local frame, we have k R(@i;@j)@l = Rlij@k: m Using the Riemannian metric, we can define a new curvature tensor Rklij := gkmRlij. Then it is easy to see Rklij = hR(@i;@j)@l;@ki: 3 The second Bianchi identity in the tangent bundle case is Rklij;h + Rlhij;k + Rhkij;l = 0; 8k; l; i; j; h: i j l k It is easy to verify Riem := Rklijdx ⊗ dx ⊗ dx ⊗ dx is a tensor field of type (0; 4) on M, called the Riemannian curvature tensor with respect to the metric g. We now define the associated intrinsic curvatures on (M; g). 1. The sectional curvature of the plane spanned by the linearly independent tangent i j vectors X = a @ijp;Y = b @jjp 2 TpM is hR(X; Y )Y; Xi K(X ^ Y ) := jX ^ Y j2 i j k l hR(a @i; b @j)b @k; a @li = i k j l i j k l gikgjl(a a b b − a a b b ) i j l k Rlkija b a b = i k j l i j k l gikgjl(a a b b − a a b b ) i j k l Rijkla b a b = i j k l ; (gikgjl − gijgkl)a b a b where jX ^ Y j2 := hX; Xi · hY; Y i − hX; Y i2. P i 2. The Ricci curvature in the direction X = i a jp 2 TpM is defined as jl Ric(X; X) := g hR(X; @j)@l;Xi jl i k = g hR(a @i;@j)@l; a @ki i k jl = a a g Rklij jl jl Hence Ric(@i;@i) = g Rilij. The Ricci tensor is Ricik := R(@i;@k) = g Rklij = jl i j g Rijkl. That is the Ricci tensor is Ric = Rikdx ⊗dx , which is of type (0; 2), while the Ricci curvature in a certain direction or along a vector field is a number at each jl point, hence is a smooth function on the manifold. We have Ricki = g Rkjil = jl lj g Rilkj = g Rilkj = Rik. Thus the Ricci tensor is a symmetric 2-tensor. ik 3. The scalar curvature is R := g Ricik. Thus the Ricci curvature in the direction X is the average of the sectional curvature of all planes in TpM containing X, and the scalar curvature is the average of the Ricci curvature of all unit vectors, i.e., of the sectional curvature of all planes in TpM. 4.
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