Connections, Lie Brackets, and Curvature

May 11, 2022 · mathdifferential geometry

Let \((M,g)\) be a smooth Riemannian manifold. Write \(\nabla\) for its Levi–Civita connection and \(T_pM\) for the tangent space at \(p\). The coordinate conventions are those in tangent spaces and derivations.

To differentiate a vector field, we must compare vectors based at different points. Coordinates permit a comparison, but their basis vectors also vary. A connection specifies how to account for this variation. We develop this derivative for vector fields on differentiable manifolds \(M\). The Lie bracket describes the commutator of vector fields acting on functions; curvature then describes the corresponding failure of covariant derivatives to commute.

1. Elementary Differential Topology

We first specify the bundles and tensor fields on which these operations act. The local trivializations keep track of how a vector space varies with its base point.

Definition 1. A triple \((\mathcal E, \Pi, M)\) is called a differentiable vector bundle of rank \(n\) if \(\mathcal E\) and \(M\) are differentiable manifolds, \(\Pi : \mathcal E \to M\) is a differentiable projection map, and each fibre \(\mathcal E_X \triangleq \Pi^{-1} (X)\) for \(X \in M\) carries the structure of a \(n\)-dimensional vector space. Furthermore, the following local triviality condition holds: for all \(X \in M\), there exists a neighbourhood \(U\) and a diffeomorphism \(\varphi : \Pi^{-1} (U) \to U \times \mathbb R^n\) with the property that for all \(y \in U\), \(\varphi_y \triangleq \varphi\big|_{\mathcal E_y} : \mathcal E_y \to \{y\} \times \mathbb R^n\) is a vector space isomorphism. By way of nomenclature, \((\varphi, U)\) is a bundle chart.

Definition 2. A \(p\)-times contravariant, \(q\)-times covariant tensor field on differentiable manifold \(M\) is a section of

\[\underbrace{TM \otimes \cdots \otimes TM}_{p \text{ times}} \otimes \underbrace{T^* M \otimes \cdots \otimes T^* M}_{q \text{ times}} ~.\]

Definition 3. For a finite-dimensional real vector space \(V\), let \(V^*=\operatorname{Hom}(V,\mathbb R)\). The space \(T^r_s(V)\) consists of multilinear maps

\[(V^*)^r\times V^s\longrightarrow\mathbb R.\]

It is canonically isomorphic to \(V^{\otimes r}\otimes(V^*)^{\otimes s}\), by evaluating each vector against the corresponding covector and each covector against the corresponding vector. In a basis, the coefficients on basis arguments determine a multilinear map uniquely, proving this identification. The space \(\Lambda^s(V^*)\) is the subspace of alternating \((0,s)\)-tensors; \(\Lambda^0(V^*)=\mathbb R\).

Definition 4. Let \(M\) be a differentiable manifold of dimension \(n\). Let \(\Pi : \mathcal E \to M\) be a projection map, and denote the \(\mathscr C^\infty\) vector bundle as \((\mathcal E, \Pi, M)\).

1. A section of \(\mathcal E\) is a differentiable map \(s : M \to \mathcal E\) such that \(\Pi \circ s = \mathbb 1_M\).

2. \(\Gamma (\mathcal E)\) denote the set of sections of \(\mathcal E\).

3. \(\Gamma (TM)\) denotes the set of vector fields (contravariant tensor fields of degree one) on \(M\), with \(TM = \bigsqcup_{p \in M} T_p M\) being the tangent bundle.

4. \(\Gamma (\Lambda_s M)\) denotes the set of \(s\)-forms on \(M\), with \(\Lambda_s M = \Lambda^s (T^*M)\) and \(T^*M = \bigsqcup_{p \in M} T_p^* M\) being the cotangent bundle.

5. \(\Gamma(T^r_s M)\) denotes the set of \((r,s)\)-tensor fields on \(M\), with \(T^r_s M\) being the tensor bundle \(\underbrace{TM \otimes \cdots \otimes TM}_{r \text{ times}} \otimes \underbrace{T^* M \otimes \cdots \otimes T^* M}_{s \text{ times}}\).

Example 5. \(g_{ij}\) is a \((0,2)\)-tensor field, i.e. \(g \in \Gamma(T^0_2 M)\) by definition.

Definition 6. A smooth symmetric \((0,2)\)-tensor field \(g\in\Gamma(T^0_2M)\) is a pseudo-Riemannian metric when each bilinear form \(g_p\) is nondegenerate: if \(g_p(v,w)=0\) for every \(v\in T_pM\), then \(w=0\). It is a Riemannian metric if each \(g_p\) is positive definite.

Definition 7. A Lorentzian metric is a smooth pseudo-Riemannian metric of index one: at each point, the maximal dimension of a subspace on which it is negative definite is one. Smoothness, rather than mere continuity, is assumed for the connection and curvature constructions below.

Remark 8. An equivalent formulation of a Lorentzian metric is the following. Quadratic form \(g_p\) in \(T_p M\) is Lorentzian if there exists \(V \in T_p M\) such that \(g_p (V,V) < 0\), while \(g_p|_{\Sigma_V}\) is a Riemannian metric, where \(\Sigma_V \triangleq \{X \in T_p M : g_p (X,V) = 0 \}\).

To verify the equivalence, choose a negative vector \(V\). Nondegeneracy and \(g(V,V)\ne0\) give \(T_pM=\mathbb RV\oplus V^\perp\). The restriction to \(V^\perp\) is nondegenerate: a vector orthogonal to both summands is orthogonal to all of \(T_pM\) and hence zero. Index one rules out a negative direction in \(V^\perp\), so its restriction is positive definite. Conversely a negative line orthogonal to a positive-definite complement has exactly one negative direction, by this direct sum decomposition.

Definition 9 (Pullback bundle). If \(f:M\to N\) is smooth and \(\pi:E\to N\) is a rank-\(r\) smooth vector bundle, define

\[f^*E=\{(p,e)\in M\times E:\pi(e)=f(p)\},\qquad\pi_f(p,e)=p.\]

Its fibre at \(p\) is the vector space \(E_{f(p)}\). If a bundle chart has the form \(\varphi(e)=(\pi(e),\xi(e))\) over \(U\subset N\), the induced chart is

\[(p,e)\longmapsto(p,\xi(e))\in f^{-1}(U)\times\mathbb R^r.\]

On overlaps the fibre-coordinate change is the original transition matrix evaluated at \(f(p)\), hence is smooth and linear. These charts define the smooth vector-bundle structure. The expression \(\varphi\circ f\) alone is ill-typed, since \(f\) takes values in the base, not in \(E\).

Definition 10. Let \((\mathcal E_1, \Pi_1, N)\) and \((\mathcal E_2, \Pi_2, N)\) denote vector bundles over \(N\). A differentiable map \(f : \mathcal E_1 \to \mathcal E_2\) is said to be fibre-preserving if \(\Pi_2 \circ f = \Pi_1\). Furthermore, let \(\mathcal E_{1,x}\) and \(\mathcal E_{2,x}\) denote the fibres of \(\mathcal E_1\) and \(\mathcal E_2\), respectively, and let \(f_x : \mathcal E_{1,x} \to \mathcal E_{2,x}\) be a linear map. Then, \(f\) is said to be a bundle homomorphism.

Definition 11. Let \((\mathcal E, \Pi, N)\) be a vector bundle of rank \(n\). Let \(\mathcal E' \subset \mathcal E\). Assume that for all \(x \in N\), there exists a bundle chart \((\varphi, U)\) such that \(x \in U\), and \(\varphi(\Pi^{-1}(U) \cap \mathcal E') = U\times \mathbb R^m \subset U \times \mathbb R^n\). Then, the restriction of the full vector bundle to \(\mathcal E'\), denoted \((\mathcal E', \Pi|_{\mathcal E'}, N)\), is called the subbundle of rank \(m\).

The details of these concepts can be found in Appendix Vector Fields and Their Flows.

2. Lie Bracket

Composing two directional derivatives usually produces a second-order operator. Taking their commutator cancels the second-order terms and leaves another vector field. A reminder of vector fields is contained in Appendix Vector Fields and Their Flows.

Let \(X, Y\) be smooth vector fields on a smooth manifold \(M\). Given a smooth real-valued function \(f \in \mathscr C^\infty (M)\), we can apply \(X\) to \(f\) and obtain another smooth function, \(Xf\). In turn, we can apply \(Y\) to this function, and obtain yet another smooth function \(YX f = Y(Xf)\). But, \(f \mapsto YX f\) is generally not a vector field. An example is \(X = \pd{}{x}\) and \(Y = x \pd{}{y}\), \(f(x,y)=x\) and \(g(x,y)=y\). Then, \(XY(fg) = X(x^2) = 2x\), but \(f XYg + gXY f = x\), so the Leibniz rule does not hold. The following object, however, is a smooth vector field.

Definition 12. Let \(M\) be a differentiable manifold. For any smooth vector fields \(X\) and \(Y\) along \(M\), the Lie bracket of \(X\) and \(Y\) is a map \([\cdot,\cdot]\) on \(\mathscr X (M) \times \mathscr X(M)\), defined as

\[[X,Y](f) = X(Yf) - Y(Xf) \text{ for all } f \in \mathscr C^\infty (M) ~.\]

Proposition 13. Let \(M\) be a differentiable manifold, and \(X,Y \in \mathscr X(M)\). Then, \([X,Y]\) is a vector field.

The Leibniz calculation below proves that the bracket is a derivation. Its smoothness is also explicit: in a chart, the second derivatives of the test function cancel, leaving coefficients \(X^i\partial_iY^j-Y^i\partial_iX^j\), which are smooth. Thus the derivation corresponds to a smooth tangent-vector field.

Proof. Linearity follows from the definition. The product rule is the substantive check: the two mixed terms must cancel. For any \(f,g \in \mathscr C^\infty (M)\),

\[\begin{aligned} [X,Y](fg) &= XY (fg) - YX(fg) = X(f Yg + g Yf) - Y(f Xg + g Xf) \\ &= fXY g + Xf Yg + gX Yf + Xg Yf - fY Xg - Yf Xg - Yg Xf - gY Xf \\ &= f[X,Y]g + g[X,Y]f ~, \end{aligned}\]

as desired. ◻

The Lie bracket also admits representation in local coordinates. In particular, let \(\{x^i\}\) be local coordinate functions for \(M\). Then, \(X = X^i \pd{}{{x^i}}\) and \(Y = Y^j \pd{}{{x^j}}\), so we can write

\[\begin{aligned} [X,Y] = \left( X^i \dpd{{Y^j}}{{x^i}} - Y^i \dpd{{X^j}}{{x^i}} \right) \dpd{}{{x^j}} ~. \end{aligned}\]

Proposition 14 (Properties of the Lie Bracket). For all \(X,Y,Z \in \mathscr X(M)\),

1. For all \(a,b \in \mathbb R\),

   \[\begin{aligned} [aX + bY, Z] &= a[X,Z] + b[Y,Z] ~, \\ [X, aY + bZ] &= a[X,Y] + b[X,Z] ~. \end{aligned}\]

2. (Skew-Symmetry) \([X,Y] = -[Y,X]\).

3. (Jacobi’s Identity) \([X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0\).

4. For all \(f,g \in \mathscr C^\infty (M)\),

   \[\begin{aligned} [fX, gY] = fg[X,Y] + (fXg) Y - (gYf) X ~. \end{aligned}\]

Proof. (a) and (b) follow from definition. For (c), let \(f \in \mathscr C^\infty (M)\). Direct calculations show that

\[\begin{aligned} [X,[Y,Z]](f) &= [X,YZ-ZY](f) = [X,YZ](f) - [X,ZY](f) \\ &= XYZf - YZXf - XZYf + ZYXf ~. \end{aligned}\]

Analogously,

\[\begin{aligned} [Y,[Z,X]](f) &= YZXf - ZXYf - YXZf + XZYf ~, \\ [Z,[X,Y]](f) &= ZXYf - XYZf - ZYXf + YXZf ~. \end{aligned}\]

All terms cancel out upon adding these three identities. For (d), let \(h \in \mathscr C^\infty (M)\). Then, direct calculations reveal that

\[\begin{aligned} [fX, gY](h) &= (fXgY)(h) - (gYfX)(h) \\ &= fg XY(h) + f(Xg)Y (h) - gf YX (h) - g(Yf) X(h) \\ &= fg[X,Y](h) + (fXg)Y (h) - (gYf) X(h) ~, \end{aligned}\]

as desired. ◻

Proposition 15. Let \(F:M \to N\) be a smooth map between differentiable manifolds \(M\) and \(N\). Let \(X_i \in \mathscr X(M)\) and \(Y_i \in \mathscr X(N)\) for \(i=1,2\). Suppose \(X_i\) is \(F\)-related to \(Y_i\) for \(i=1,2\). Then, \([X_1,X_2]\) is \(F\)-related to \([Y_1,Y_2]\).

Proof. By Proposition 7, and by the assumption that \(X_i\) and \(Y_i\) are \(F\)-related, we have

\[X_1 X_2 (f \circ F) = X_1 ((Y_2 f) \circ F) = (Y_1 Y_2 f) \circ F ~.\]

Meanwhile, we have

\[X_2 X_1 (f \circ F) = (Y_2 Y_1 f) \circ F ~.\]

Collecting these terms, we have

\[\begin{aligned} [X_1, X_2](f \circ F) &= X_1 X_2 (f \circ F) - X_2 X_1 (f \circ F) = (Y_1 Y_2 f) \circ F - (Y_2 Y_1 f) \circ F \\ &= ([Y_1, Y_2] f) \circ F ~, \end{aligned}\]

concluding the proof. ◻

Corollary 16.

1. Let \(F: M \to N\) be a diffeomorphism between differentiable manifolds \(M\) and \(N\). Let \(X_1, X_2 \in \mathscr X(M)\). Then, \(F_*[X_1,X_2] = [F_* X_1, F_* X_2]\).

2. Suppose \(S\) is an immersed submanifold of \(M\). If \(Y_1\) and \(Y_2\) are smooth vector fields on \(M\) tangent to \(S\), then \([Y_1,Y_2]\) is also tangent to \(S\).

Proof. For (a), the pushforward \(F_*X\) is smooth because \(F\) is a diffeomorphism and is, by definition, \(F\)-related to \(X\). Proposition 15 gives the asserted equality.

For (b), write the immersion as \(\iota:S\to M\), and interpret tangency branchwise: \(Y_i(\iota(s))\in d\iota_s(T_sS)\) for every \(s\in S\). Since \(d\iota_s\) is injective, this specifies a unique lifted field \(X_i(s)\). It is smooth, as seen in an immersion chart where \(\iota(u)=(u,0)\) and the tangential components of \(Y_i\) restrict smoothly to \(S\). Proposition 15 now gives \(d\iota[X_1,X_2]=[Y_1,Y_2]\circ\iota\), proving tangency even when the immersed image has more than one local branch. ◻

3. Affine and Riemannian Connection

A connection extends directional differentiation to vector fields. Its two product rules distinguish the direction of differentiation from the field being differentiated.

Definition 17. Let \(M\) be a differentiable manifold. An affine connection on \(M\), denoted \(\nabla\), is a bilinear map

\[ \begin{aligned} \nabla : \Gamma(TM) \times \Gamma(TM) &\to \Gamma(TM) \\ (X,Y) &\mapsto \nabla_X Y \end{aligned}\tag{1}\]

such that

1. \(\nabla_{fX} Y = f \nabla_X Y\)

2. \(\nabla_X (fY) = f \nabla_X Y + X(f) Y\)

for vector fields \(X,Y \in \Gamma(TM)\) and \(f \in \mathscr C^{\infty} (M)\).

Definition 18. Given an affine connection \(\nabla\),

1. The map \(T : \Gamma(M) \times \Gamma(M) \to \Gamma(M)\), defined \(T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]\), is called the torsion tensor of \(\nabla\).

2. \(\nabla\) is torsion-free if \(T(X,Y) = 0\) for all \(X, Y \in \Gamma(M)\).

3. If \(g\) is a Riemannian metric on manifold \(M\), \(\nabla\) is called metric with respect to \(g\) if the following identity holds:

   \[\begin{aligned} Z g(X,Y) = g(\nabla_Z X, Y) + g(X, \nabla_Z Y) \quad \forall X,Y,Z \in \Gamma(TM) ~. \end{aligned}\]

There is exactly one connection that is torsion-free and metric with respect to the Riemannian metric \(g\) on a differentiable manifold \(M\).

Theorem 19 (Levi-Civita). On each Riemannian manifold \((M,g)\), there exists a unique metric, torsion-free connection. We call this the Riemannian connection or Levi-Civita connection. Furthermore, this unique connection is characterized through the identity

\[ \begin{aligned} \langle \nabla_X Y , Z \rangle &= \frac12 \left( X \langle Y, Z \rangle - Z \langle X, Y \rangle + Y \langle Z , X \rangle - \langle X, [Y,Z] \rangle + \langle Z, [X,Y] \rangle + \langle Y, [Z,X] \rangle \right) ~. \end{aligned}\tag{2}\]

(2) is called the Koszul formula.

Proof of existence and uniqueness. Metric compatibility gives the three identities for \(Xg(Y,Z)\), \(Yg(Z,X)\), and \(Zg(X,Y)\). Add the first two and subtract the third. Use torsion-freeness to replace \(\nabla_YX\) by \(\nabla_XY-[X,Y]\), and similarly for the other two interchanges. Collecting terms yields exactly twice \(g(\nabla_XY,Z)\), which is the displayed Koszul formula. Since the metric is nondegenerate, it determines \(\nabla_XY\) uniquely.

For existence, call the right-hand side \(K(X,Y,Z)\). It is \(C^\infty(M)\)-linear in \(Z\): expanding \(K(X,Y,fZ)\) by the product rule and \([X,fZ]=f[X,Z]+(Xf)Z\) makes every derivative-of-\(f\) term cancel. Thus \(Z\mapsto K(X,Y,Z)\) is a smooth one-form, and the inverse metric gives a unique smooth vector field \(\nabla_XY\). Expanding the same formula gives

\[K(fX,Y,Z)=fK(X,Y,Z),\quad K(X,fY,Z)=fK(X,Y,Z)+(Xf)g(Y,Z).\]

It is real-bilinear in \(X,Y\), so these identities are the connection rules. Subtracting the formula with \(X,Y\) exchanged gives \(K(X,Y,Z)-K(Y,X,Z)=g([X,Y],Z)\), proving torsion-freeness. Adding the formulas for \(K(X,Y,Z)\) and \(K(X,Z,Y)\) gives \(Xg(Y,Z)\), proving metric compatibility. In coordinate fields the brackets vanish; solving for coefficients recovers

\[\nabla_{\partial_i}\partial_j=\Gamma^k_{ij}\partial_k,\qquad \Gamma^k_{ij}=\frac12g^{k\ell}(\partial_i g_{j\ell}+\partial_jg_{i\ell}-\partial_\ell g_{ij}).\]

This constructs the connection and checks every required property. ◻

Through Theorem 19, we can do two things. One, we can define a geodesic through the Levi-Civita connection \(\nabla\).

Proposition 20. \(\gamma\) is a geodesic on \(M\) if \(\nabla_{\dot \gamma} \dot \gamma = 0\), where \(\nabla\) is the Levi-Civita connection on \(M\).

Proof. For a vector field \(V=V^k(t)\partial_k\) along \(\gamma\), the induced covariant derivative is

\[D_tV=\left(\dot V^k+\Gamma^k_{ij}(\gamma(t))\dot\gamma^i V^j\right)\partial_k.\]

The derivative of the component is a time derivative; there is no extra factor \(\dot\gamma^i\) in front of it. Taking \(V=\dot\gamma\) yields

\[D_t\dot\gamma=(\ddot\gamma^k+\Gamma^k_{ij}\dot\gamma^i\dot\gamma^j)\partial_k.\]

The coordinate basis is linearly independent, so this is zero exactly when each geodesic-equation component is zero. Hence the equivalence holds in both directions. ◻

4. Riemann Curvature

The connection also gives the curvature tensor. The correction by the Lie bracket makes the commutator of covariant derivatives tensorial, so its value at a point depends only on the tangent vectors there.

Definition 21 (Curvature convention). We use

\[R(X,Y)Z=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z.\tag{3}\]

This is the negative of the convention with \(\nabla_X\nabla_Y\) first. The plus sign before the bracket is essential with the order chosen here: it is precisely what makes the expression tensorial. With this convention, sectional curvature is \(g(R(X,Y)X,Y)\) for an orthonormal pair.

Proposition 22. The Riemann curvature on a differentiable manifold satisfies the following properties:

1. Bilinearity: for all smooth functions \(f,g\) on \(M\) and \(X,Y,Z \in \Gamma(M)\), we have

   \[\begin{aligned} R(fX + gY, Z) &= f R(X,Z) + g R(Y,Z) \\ R(X, fY + gZ) &= f R(X,Y) + g R(X,Z) \end{aligned}\]

2. Linearity of the curvature operator: for all smooth function \(f\) on \(M\) and \(X,Y,Z,W \in \Gamma(M)\), we have

   \[\begin{aligned} R(X,Y) (Z+W) &= R(X,Y) Z + R(X,Y) W \\ R(X,Y) fZ &= f R(X,Y) Z \end{aligned}\]

Proof. Additivity follows from the connection rules. For multiplication in the first slot, use \([fX,Y]=f[X,Y]-(Yf)X\). The term \((Yf)\nabla_XZ\) from \(\nabla_Y(f\nabla_XZ)\) cancels the corresponding term from the bracket. Hence \(R(fX,Y)Z=fR(X,Y)Z\). Skew-symmetry in \(X,Y\), immediate from Definition 21, gives the second-slot statement.

For the third slot, expansion gives

\[R(X,Y)(fZ)=fR(X,Y)Z+\{Y(Xf)-X(Yf)+[X,Y]f\}Z.\]

The mixed terms \((Xf)\nabla_YZ\) and \((Yf)\nabla_XZ\) cancel before this expression is obtained. The scalar coefficient shown is zero by the definition of the Lie bracket. Therefore \(R\) is \(C^\infty\)-linear in all three slots and defines a \((1,3)\)-tensor. ◻

Since \(R\) is tensorial, it makes sense to describe it through (any) coordinates. In particular, consider the coordinate vector fields \(X = \dpd{}{{x^i}}\), \(Y = \dpd{}{{x^j}}\), and \(Z = \dpd{}{{x^{k}}}\). This gives us a description of \(R\) in coordinates, which we denote \(R_{k\ell ij}\).

Proposition 23 (Coordinate conventions). Write

\[R(\partial_i,\partial_j)\partial_k=R^m_{ijk}\partial_m,\qquad R_{k\ell ij}:=g(R(\partial_i,\partial_j)\partial_\ell,\partial_k) =g_{km}R^m_{ij\ell}.\tag{4}\]

Then

\[R^m_{ijk}=\partial_j\Gamma^m_{ik}-\partial_i\Gamma^m_{jk} +\Gamma^p_{ik}\Gamma^m_{jp}-\Gamma^p_{jk}\Gamma^m_{ip}.\tag{5}\]

The four-index order is fixed by the scalar-product definition, rather than inferred from a permutation of three-index subscripts.

Proof. Let \(X = \dpd{}{{x^i}}\), \(Y = \dpd{}{{x^j}}\), and \(Z = \dpd{}{{x^{k}}}\). Since \(X,Y\) are coordinate vector fields, \([X,Y] =0\). It remains to calculate the first two terms of (3). Clearly,

\[\begin{aligned} \nabla_{{\partial}/{\partial{x^j}}} \nabla_{{\partial}/{\partial{x^i}}} \dpd{}{{x^{k}}} - \nabla_{{\partial}/{\partial{x^i}}} \nabla_{{\partial}/{\partial{x^j}}} \dpd{}{{x^{k}}} &= \nabla_{{\partial}/{\partial{x^j}}} \Gamma_{ik}^{m} \dpd{}{{x^m}} - \nabla_{{\partial}/{\partial{x^i}}} \Gamma_{jk}^{m} \dpd{}{{x^m}} \\ &= \Gamma^p_{ik} \Gamma^m_{jp} \dpd{}{{x^m}} + \dpd{\Gamma_{ik}^m}{{x^j}} \dpd{}{{x^m}} - \left( \Gamma^p_{jk} \Gamma^m_{ip} \dpd{}{{x^m}} + \dpd{\Gamma_{jk}^m}{{x^i}} \dpd{}{{x^m}} \right) ~, \end{aligned}\]

and equating this with the left-hand side of (4) gives the desired result. ◻

Proposition 24 (Curvature symmetries). For the Levi–Civita connection,

\[R(X,Y)Z=-R(Y,X)Z,\]

\[R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0,\]

\[g(R(X,Y)Z,W)=-g(R(X,Y)W,Z),\]

\[g(R(X,Y)Z,W)=g(R(Z,W)X,Y).\]

In the four-index convention of Proposition 23, these read

\[R_{k\ell ij}=-R_{k\ell ji},\quad R_{k\ell ij}+R_{ki j\ell}+R_{kj\ell i}=0,\quad R_{k\ell ij}=-R_{\ell kij},\quad R_{k\ell ij}=R_{ijk\ell}.\]

Proof. First, interchanging \(X,Y\) in Definition 21 changes the sign of both the commutator and the bracket. For the first Bianchi identity it suffices, by tensoriality, to take coordinate fields at a point. Insert formula (5) and sum cyclically in \(i,j,k\). The derivative terms cancel in pairs because \(\Gamma^m_{ij}=\Gamma^m_{ji}\). The product terms cancel too: for instance \(\Gamma^p_{ik}\Gamma^m_{jp}\) cancels \(-\Gamma^p_{ki}\Gamma^m_{jp}\) from the next cyclic term. This proves the cyclic identity without any assumption that the fields commute away from that coordinate choice.

For skew-adjointness, apply the scalar operator \(YX-XY+[X,Y]\), which is zero, to \(g(Z,W)\). Metric compatibility expands this into

\[0=g(R(X,Y)Z,W)+g(Z,R(X,Y)W);\]

the terms involving one derivative on each of \(Z,W\) cancel between the two orders. This gives the third symmetry.

For pair exchange, a direct normal-coordinate computation avoids an implicit permutation argument. At a chosen point \(p\), \(g_{ij}=\delta_{ij}\), \(\partial_k g_{ij}=0\), and \(\Gamma^m_{ij}=0\). Insert the Christoffel formula into (5), then lower the output index according to (4). Cancellation of the \(g_{ij,k\ell}\) terms gives

\[R_{ijk\ell}(p)=\frac12\bigl(g_{ik,j\ell}-g_{jk,i\ell}-g_{i\ell,jk}+g_{j\ell,ik}\bigr)(p).\tag{6}\]

Here commas denote ordinary partial derivatives. Exchanging the index pairs \((i,j)\) and \((k,\ell)\) leaves this expression unchanged, using symmetry of the metric and commutation of its mixed partials. Thus the pair-exchange identity holds at \(p\). Since both sides are tensors and \(p\) was arbitrary, it holds everywhere in every chart. This computation also independently verifies the minus signs in both skew symmetries. ◻

Proposition 25 (Second Bianchi identity). The covariant derivative of the curvature tensor satisfies

\[(\nabla_XR)(Y,Z)W+(\nabla_YR)(Z,X)W+(\nabla_ZR)(X,Y)W=0.\]

The parentheses mean the tensor derivative:

\[(\nabla_XR)(Y,Z)W=\nabla_X(R(Y,Z)W)-R(\nabla_XY,Z)W-R(Y,\nabla_XZ)W-R(Y,Z)\nabla_XW.\]

Without these correction terms the asserted identity is generally false for varying vector fields.

Proof. The tensor-derivative expression is tensorial in all four arguments, so fix a point \(p\) and choose normal coordinates there. Use coordinate vectors for \(X,Y,Z,W\). At \(p\), covariant differentiation of the components is ordinary partial differentiation, since all Christoffel coefficients vanish. Differentiate the general formula (5), not an identity known only at a single point. A derivative of either quadratic Christoffel term contains an undifferentiated Christoffel factor and therefore vanishes at \(p\). Thus

\[\partial_a R^m_{bc d}(p)=\partial_a\partial_c\Gamma^m_{bd}(p)-\partial_a\partial_b\Gamma^m_{cd}(p).\]

Adding the corresponding expressions with \((a,b,c)\) replaced by \((b,c,a)\) and \((c,a,b)\) cancels all six terms by equality of mixed partial derivatives. This proves the required cyclic sum at \(p\), and tensoriality proves it on the manifold. ◻

Lemma 26 (Commuting derivatives along a variation). Let \(F:(-\varepsilon,\varepsilon)\times[a,b]\to M\) be smooth, put \(S=\partial_sF\), \(T=\partial_tF\), and let \(V\) be a smooth vector field along \(F\). Denoting the pullback covariant derivatives by \(D_s,D_t\), our curvature convention gives

\[D_tD_sV-D_sD_tV=R(S,T)V.\]

No immersion assumption on \(F\) is needed.

Proof. In ambient coordinates \(x^i\), write \(F^i=x^i\circ F\) and \(V=v^k\partial_k\). Then

\[(D_sv)^m=\partial_sv^m+\Gamma^m_{ik}(F)\,\partial_sF^i\,v^k,\]

and similarly for \(D_t\). Expand \(D_tD_sV-D_sD_tV\). Mixed second derivatives of \(v\) cancel; the terms involving first derivatives of \(v\) also cancel; and the coefficients of \(\partial_{st}F^i-\partial_{ts}F^i\) are zero. The remaining component is

\[\partial_sF^i\,\partial_tF^j\,v^k \bigl(\partial_j\Gamma^m_{ik}-\partial_i\Gamma^m_{jk} +\Gamma^p_{ik}\Gamma^m_{jp}-\Gamma^p_{jk}\Gamma^m_{ip}\bigr).\]

By (5) this is the \(m\)-th component of \(R(S,T)V\). This coordinate proof takes derivatives of the pullback bundle and never treats the parameter derivatives as independent ambient coordinate fields. ◻

Remark 27. For the covariant derivative along a curve \(c:[a,b]\to M\), it is frequently the case that texts write \(\frac{\Dif}{\Dif t}\) in place of \(\nabla_{\dif c/\!\dif t}\). Wherever the context is clear, we write \(\nabla_{\dif/\!\dif t}\) for covariant derivative along curve \(c\) parametrized by \(t\) only. By analogy, in the sequel, we write \(\nabla_{\partial/\partial t}\) for covariant derivatives “in the \(t\) direction”, along variations parametrized by \((s,t)\).

Remark 28 (Normal-coordinate Taylor terms). The later Jacobi-field calculation gives, in the four-index convention (4),

\[g_{ij}(x)=\delta_{ij}-\frac13R_{ik\ell j}(p)x^kx^\ell+O(|x|^3).\]

The coefficients in this polynomial are evaluated at \(p\). Taylor's theorem for the smooth metric, including its differentiated remainders, gives

\[\partial_r g_{ij}(x)=-\frac13(R_{ir\ell j}(p)x^\ell+R_{ikrj}(p)x^k)+O(|x|^2),\]

\[\partial_s\partial_rg_{ij}(x)=-\frac13(R_{irsj}(p)+R_{isrj}(p))+O(|x|).\]

The minus signs persist under differentiation, and the remainder order drops with each derivative. This conclusion uses a smooth Taylor remainder; differentiating an arbitrary big-O bound alone would not justify it. The next lemma is proved directly from the connection, so no circular use of this later Taylor expansion is needed.

Lemma 29. In normal coordinates centred at \(p\), with \(R_{ijk\ell}=g(R(\partial_k,\partial_\ell)\partial_j,\partial_i)\),

\[R_{ijk\ell}(p)=\frac12\left(\partial_j\partial_\ell g_{ik}-\partial_i\partial_\ell g_{jk}-\partial_j\partial_k g_{i\ell}+\partial_i\partial_k g_{j\ell}\right)(0).\tag{7}\]

Proof. At the origin, the quadratic Christoffel terms vanish and \(g_{im}=\delta_{im}\), so

\[R_{ijk\ell}=\partial_\ell\Gamma^i_{kj}-\partial_k\Gamma^i_{\ell j}.\]

Since \(\partial g^{im}(0)=0\), differentiating the Christoffel formula at zero gives

\[\partial_\ell\Gamma^i_{kj}=\frac12(g_{ji,k\ell}+g_{ki,j\ell}-g_{kj,i\ell}),\quad \partial_k\Gamma^i_{\ell j}=\frac12(g_{ji,\ell k}+g_{\ell i,jk}-g_{\ell j,ik}).\]

Subtract; the first terms cancel by equality of mixed partials, leaving (7). For a unit sphere, this convention gives \(R_{1221}=1\), consistent with positive sectional curvature and the negative quadratic normal-coordinate metric term. ◻

Reading: Carmo (1992), Riemannian geometry.

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