The Involution on the Curvature Operator

Introduction

The curvature tensor of a Riemannian manifold is a tensor of four arguments with a rigid pattern of symmetries, and the pattern is best described by two involutions. The first is the skew-symmetry: the tensor is alternating in the first pair of arguments and in the second pair separately, so it is a two-form in each pair and lives in $\Lambda^2\otimes\Lambda^2$. The second is the pair symmetry: the tensor is unchanged when the two pairs are exchanged, and this is exactly the statement that the curvature operator $\mathcal{R}$ of The Curvature Operator is a self-adjoint operator on the space of the bivectors $\Lambda^2$. The two involutions, together with the first Bianchi identity, characterise the curvature tensors among all the symmetric bilinear forms on the bivectors. A third involution enters in even dimension, the Hodge star on $\Lambda^2$, and its $\pm1$-eigenspaces decompose the curvature operator into the self-dual and the anti-self-dual parts, the decomposition that organises the curvature of a four-manifold.

The article develops the involutions that the curvature tensor carries. It reads the skew-symmetry of the curvature as the statement that the tensor is a section of $\Lambda^2\otimes\Lambda^2$; it proves that the pair symmetry is the self-adjointness of the curvature operator and that the first Bianchi identity is the vanishing of the Bianchi map, so that the curvature tensors are the self-adjoint operators lying in the kernel of the Bianchi map; it proves that the curvature operator is diagonalisable with real eigenvalues and that the sectional curvature is the quadratic form it defines; it develops the Hodge star as an involution of $\Lambda^2$ in even dimension and the self-dual and anti-self-dual splitting it induces on the curvature operator; and it shows how an isometric involution, a real structure and a complex structure act on the curvature operator by an involution that preserves its symmetries.

The article assumes the curvature, the curvature tensor and the sectional curvature of Curvature and Geodesics and Riemannian Geometry, the curvature operator $\mathcal R$ on the bivectors and its expression in an orthonormal frame from The Curvature Operator, and the isometric involutions from the preceding articles of this category; the Hodge star and the exterior algebra are Part III's. The Einstein condition, the Ricci flow and the four-dimensional geometry are Ricci Flow and Curvature and Geodesics elsewhere in this Part, and the Kähler geometry is the Hermitian article of this group. No physics is invoked.

The Symmetries of the Curvature Tensor

The Skew-Symmetry

Definition. The curvature tensor of a Riemannian manifold is the tensor

$$ R(X, Y, Z, W) = \bigl\langle R(X, Y)Z,\, W\bigr\rangle, $$

and it is alternating in each pair: for all vectors $X, Y, Z, W$,

$$ R(X, Y, Z, W) = -R(Y, X, Z, W), \qquad R(X, Y, Z, W) = -R(X, Y, W, Z). $$

Equivalently, the curvature is a section of $\Lambda^2T^*M\otimes\Lambda^2T^*M$: it is a two-form in the pair $(X, Y)$ and a two-form in the pair $(Z, W)$.

Proof. The first identity is $R(X,Y) = -R(Y,X)$, which is the antisymmetry of the curvature operator of The Curvature Operator, proved from the antisymmetry of the Lie bracket and the connection. The second is the metricity of the connection read on the curvature, $R(X,Y,Z,W) = -R(X,Y,W,Z)$, which follows because $R(X,Y)$ is a skew-adjoint endomorphism of the tangent space for the metric; the skew-adjointness is the identity $\langle R(X,Y)Z,W\rangle = -\langle Z,R(X,Y)W\rangle$, which is the definition of the curvature of the Levi-Civita connection as a metric connection.

The Pair Symmetry and the Bianchi Identity

Theorem. The curvature tensor has the pair symmetry

$$ R(X, Y, Z, W) = R(Z, W, X, Y), $$

and satisfies the first Bianchi identity

$$ R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0, $$

equivalently $R(X,Y,Z,W) + R(Y,Z,X,W) + R(Z,X,Y,W) = 0$. The pair symmetry is the statement that the curvature operator $\mathcal{R} : \Lambda^2 T_pM \to \Lambda^2 T_pM$ is self-adjoint for the induced metric on the bivectors, and the Bianchi identity is the statement that the value of $\mathcal{R}$ lies in the kernel of the Bianchi map $b : \Lambda^2\otimes\Lambda^2 \to \Lambda^4$.

Proof. The pair symmetry is the classical identity of the curvature tensor, proved from the symmetry of the second covariant derivatives or, equivalently, from the fact that the holonomy argument of The Parallel Transport Operator makes the curvature the bracket of two connection forms and the pairing symmetric. In the bivector language it is the self-adjointness of $\mathcal{R}$: the identity $\langle\mathcal{R}(\omega),\eta\rangle = \langle\omega,\mathcal{R}(\eta)\rangle$ for the bivectors $\omega = X\wedge Y$ and $\eta = Z\wedge W$ is exactly the pair symmetry. The Bianchi identity is the Jacobi identity of the Lie bracket transported to the curvature, $R(X,Y)Z = [\nabla_X,\nabla_Y]Z - \nabla_{[X,Y]}Z$, together with the vanishing of the curvature of the flat connection of the coordinate frame; the identification with the kernel of the Bianchi map is the definition of the map.

The Algebraic Curvature Tensors

Definition. An algebraic curvature tensor at a point is a tensor with the symmetries of the curvature: alternating in each pair, with the pair symmetry, and satisfying the first Bianchi identity. Equivalently, it is a self-adjoint operator $\mathcal{R} : \Lambda^2 \to \Lambda^2$ lying in the kernel of the Bianchi map, $\mathcal{R}\in S^2(\Lambda^2)\cap\ker b$.

Theorem. The space of the algebraic curvature tensors is exactly the space $S^2(\Lambda^2)\cap\ker b$ of the self-adjoint operators lying in the kernel of the Bianchi map, and the Ricci contraction maps it onto the space of the symmetric two-tensors; the kernel of the contraction is the space of the Weyl tensors, and the orthogonal decomposition of the algebraic curvature tensor into the Ricci part and the Weyl part is the orthogonal decomposition of this space under the action of the orthogonal group. The Bianchi identity is the only linear relation imposed on an otherwise arbitrary self-adjoint operator on the bivectors, so the algebraic curvature tensors are a linear subspace of $S^2(\Lambda^2)$ defined by it.

Proof sketch. The symmetries are those of the previous two theorems, which identify the algebraic curvature tensors with the self-adjoint operators satisfying the Bianchi identity; the Ricci contraction is the contraction of the tensor in the first and last arguments and is equivariant for the orthogonal group, its kernel being the trace-free part of the curvature, which is the Weyl tensor; the decomposition follows from the equivariance and the orthogonal splitting of the space of the tensors. The details of the Ricci–Weyl decomposition are in The Curvature Operator, Curvature and Geodesics and the references.

The Curvature Operator

Self-Adjointness

Theorem. The curvature operator is self-adjoint for the metric on the bivectors,

$$ \mathcal{R}^{*} = \mathcal{R}, \qquad \langle\mathcal{R}(\omega),\eta\rangle = \langle\omega,\mathcal{R}(\eta)\rangle \quad (\omega,\eta\in\Lambda^2T_pM), $$

so it is diagonalisable with real eigenvalues and an orthonormal eigenbasis of the bivectors; the eigenvalues are the principal curvatures of the curvature operator, and the sectional curvature of a unit bivector is the quadratic form

$$ K(\omega) = \langle\mathcal{R}(\omega), \omega\rangle . $$

If the operator is positive semidefinite then all the sectional curvatures are nonnegative, and if it is negative semidefinite then they are nonpositive; the converse fails in dimension at least four, where the decomposable bivectors form a proper cone and the curvature operator is a strictly finer invariant than the sectional curvature, as in The Curvature Operator. The Einstein condition is the statement that the Ricci contraction of $\mathcal{R}$ is a multiple of the metric.

Proof. The self-adjointness is the pair symmetry of the previous theorem; a self-adjoint operator on a finite-dimensional inner product space is diagonalisable with real eigenvalues by the spectral theorem; the sectional curvature is the quadratic form by the definition of the curvature operator and the identification of the bivectors. The sign statements follow because a semidefinite quadratic form is nonnegative, respectively nonpositive, on the decomposable cone contained in the whole space; the converse is the dimension count recorded in The Curvature Operator. The Einstein condition is the trace statement, developed in the references.

The Symmetry and the Skew-Symmetry Together

Proposition. The two involutions of the curvature are the antisymmetry and the symmetry of the same object seen in the two tensor factors: the antisymmetry is the sign of the exchange of the order inside a factor, and the pair symmetry is the exchange of the two factors. The curvature operator is the same datum as the pair-symmetric, factor-alternating tensor, and the two involutions generate the group of the symmetries of the curvature tensor.

Proof. The identification is the equivalence between the tensor $R(X,Y,Z,W)$ and the operator $\mathcal{R}$, with the metric raising and lowering the indices; the antisymmetry is the definition of $\Lambda^2$, and the pair symmetry is the self-adjointness. The group generated by the two involutions is the group of the permutations of the arguments that preserves the curvature identities.

The Hodge Involution and the Self-Dual Splitting

The Star Involution on Λ²

Definition. On an oriented Riemannian manifold of even dimension $n = 2m$ the Hodge star is the operator $\star : \Lambda^k \to \Lambda^{n-k}$ defined by $\alpha\wedge\star\beta = \langle\alpha,\beta\rangle\mathrm{vol}$, and on the bivectors it is an involution up to sign,

$$ \star^2 = (-1)^{k(n-k)}\mathrm{id} = \mathrm{id}\quad\text{on}\ \Lambda^2\ (k=2), $$

for every even $n$; it is an isometry of the exterior algebra, and it commutes with the metric. In dimension four it is an involution of $\Lambda^2$ onto itself and it splits the bivectors into the self-dual and the anti-self-dual parts,

$$ \Lambda^2 = \Lambda^2_+\oplus\Lambda^2_-, \qquad \star = +\mathrm{id}\ \text{on}\ \Lambda^2_+, \qquad \star = -\mathrm{id}\ \text{on}\ \Lambda^2_-, $$

the two eigenspaces of the involution, of dimension three each in the four-dimensional case.

Proof. The identity $\star^2 = (-1)^{k(n-k)}$ is the classical computation in an oriented orthonormal basis, and on $\Lambda^2$ in even dimension $(-1)^{2(n-2)} = +1$; the star is an isometry because the pairing defining it is the metric pairing. In dimension four $\star$ maps $\Lambda^2$ to itself and its eigenvalues $\pm1$ have the eigenspaces of dimension $\binom{4}{2}/2 = 3$; the two eigenspaces are the self-dual and the anti-self-dual bivectors.

The Self-Dual and Anti-Self-Dual Parts

Theorem. In dimension four the curvature operator commutes with the Hodge involution exactly when the curvature is self-dual or anti-self-dual in the appropriate sense; in general the operator splits into the four blocks $\Lambda^2_{\pm}\to\Lambda^2_{\pm}$ and $\Lambda^2_{\pm}\to\Lambda^2_{\mp}$, and the Weyl tensor splits into the self-dual and the anti-self-dual parts $W = W_+ + W_-$. A four-manifold is Einstein exactly when the two mixed blocks of the curvature operator vanish at every point, and it is self-dual (or anti-self-dual) when in addition the corresponding diagonal block of the Weyl part vanishes.

Proof sketch. The self-duality and the anti-self-duality are the invariance or the anti-invariance of the curvature under $\star$, which is the vanishing of the mixed blocks of the operator; the Einstein condition reduces the Ricci part to a multiple of the metric and kills the traceless Ricci, which is the vanishing of the mixed blocks; the Weyl splitting is the decomposition of the trace-free part under the star involution. The four-dimensional geometry, the self-dual metrics and the conformal geometry are Curvature and Geodesics, Ricci Flow and the references.

The Involution on the Curvature Operator

Commutation with the Isometric Involution

Theorem. Let $\sigma$ be an isometric involution of $(M, g)$, with the conjugation $\mathrm{ad}_\sigma$ on the operator layer of Isometric Involutions on the Operator Layer, the preceding article. Then the conjugation acts on the curvature tensor and on the curvature operator by the induced involutions of the tangent space,

$$ \mathrm{ad}_\sigma(R)(X,Y,Z,W) = R(d\sigma X, d\sigma Y, d\sigma Z, d\sigma W), \qquad \mathrm{ad}_\sigma(\mathcal{R}) = \mathcal{R}, $$

so the curvature operator is a fixed operator of the involution, and it commutes with the involution of the bivectors induced by $d\sigma$; the two involutions of the curvature tensor — the skew-symmetry and the pair symmetry — are preserved by $\sigma$, and the Hodge involution is preserved by an orientation-preserving $\sigma$ and conjugated by an orientation-reversing one.

Proof. The conjugation by an isometry carries the curvature to the curvature by the naturality of the curvature under an isometry, which is the formula displayed; the curvature operator is therefore fixed under the conjugation, which is the statement of Isometric Involutions on the Operator Layer. The symmetry of the curvature is preserved because $\sigma$ is a metric isometry, hence preserves the two involutions of the tensor; the Hodge involution is built from the metric and the orientation, so an orientation-preserving $\sigma$ commutes with it and an orientation-reversing one conjugates it by the sign.

The Kähler Invariance

Proposition. On a Kähler manifold the complex structure is parallel and the curvature operator commutes with the induced involution $J$ on the bivectors: the curvature tensor is $J$-invariant, $R(JX, JY) = R(X, Y)$ as an identity of endomorphisms, and the curvature operator preserves the decomposition of $\Lambda^2\otimes\mathbb{C}$ into the $(+i)$- and $(-i)$-eigenspaces of $J$; the holomorphic sectional curvature is the quadratic form of $\mathcal R$ on the bivectors of type $(1,1)$. The Hermitian and Kähler case is Hermitian Manifolds and the Geodesic Involution of this category and Hermitian Geometry and Almost Complex Structures of a later category.

Proof. The complex structure is parallel on a Kähler manifold, so it is a parallel endomorphism of the tangent bundle and it acts on the curvature tensor by the naturality of the curvature; the invariance $R(JX,JY) = R(X,Y)$ follows because $J$ is a parallel isometry of the tangent bundle. The decomposition statement is the spectral decomposition of $J$ on the complexified bivectors, and the holomorphic sectional curvature is the restriction of the quadratic form to the $(1,1)$-bivectors.

Examples

Example (the flat space). In Euclidean space the curvature tensor vanishes, the curvature operator is the zero operator, and both involutions act on the zero tensor; the Bianchi map and the star involution are the algebraic framework in which the vanishing is expressed.

Example (the space forms). For a space of constant curvature $\lambda$ the curvature tensor is $R(X,Y,Z,W) = \lambda(\langle X,Z\rangle\langle Y,W\rangle - \langle X,W\rangle\langle Y,Z\rangle)$ and the curvature operator is $\mathcal{R} = \lambda\,\mathrm{id}$ on the bivectors: it is a multiple of the identity, self-adjoint, and it commutes with every involution of the bivectors, including the Hodge star. The sphere and the hyperbolic space have $\lambda>0$ and $\lambda<0$, and the eigenvalues of $\mathcal{R}$ are all $\lambda$.

Example (the complex projective space). On $\mathbb{CP}^n$ with the Fubini–Study metric the curvature operator leaves the $(1,1)$-bivectors invariant and the holomorphic sectional curvature is the constant positive value; the curvature is not a multiple of the identity unless $n=1$, and the invariant decomposition of the bivectors into the $(1,1)$ and the $(2,0)+(0,2)$ parts is the one that organises the curvature. In dimension four the real projective space, the complex projective space and the other symmetric spaces exhibit the self-dual and the anti-self-dual splitting of the curvature and the Einstein condition.

Summary

The curvature tensor carries two involutions: the skew-symmetry, alternating in each of its two pairs of arguments, which makes it a section of $\Lambda^2\otimes\Lambda^2$, and the pair symmetry, the exchange of the two pairs, which is exactly the self-adjointness of the curvature operator $\mathcal{R}$ on the bivectors. The first Bianchi identity is the additional relation that characterises the curvature among the self-adjoint operators: the curvature tensors are the self-adjoint operators in the kernel of the Bianchi map, $\mathcal R\in S^2(\Lambda^2)\cap\ker b$, and they decompose into the Ricci and the Weyl parts under the orthogonal group.

The curvature operator is self-adjoint, hence diagonalisable with real eigenvalues, and the sectional curvature is its quadratic form $K(\omega) = \langle\mathcal R(\omega),\omega\rangle$: a positive semidefinite operator gives nonnegative sectional curvature and a negative semidefinite one gives nonpositive sectional curvature, the converse failing in dimension at least four, and the Einstein condition is the trace statement. In even dimension the Hodge star is an involution of the bivectors, $\star^2=+1$ on $\Lambda^2$, and in dimension four it splits them into the self-dual and the anti-self-dual parts $\Lambda^2_\pm$; the curvature operator splits into the four blocks, the Einstein condition is the vanishing of the mixed blocks, and the Weyl tensor splits into the self-dual and the anti-self-dual parts. An isometric involution fixes the curvature operator, and the conjugation by $\sigma$ preserves the two symmetries of the curvature and commutes with or conjugates the Hodge involution according to the orientation; on a Kähler manifold the complex structure is a parallel involution whose curvature invariance organises the curvature into the $(1,1)$ and the $(2,0)+(0,2)$ parts.

Summary of Notation

Further Reading

• Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I (Interscience, 1963), for the curvature tensor, its symmetries and the Bianchi identities.
• Manfredo P. do Carmo, Riemannian Geometry (Birkhäuser, 1992), for the curvature tensor, the sectional curvature and the algebraic structure.
• Robert Osserman, "Curvature in the eighties", American Mathematical Monthly 97 (1990), 731–756, for the curvature operator, the algebraic curvature tensors and the decomposition of the curvature.
• Arthur L. Besse, Einstein Manifolds (Springer, 1987), for the curvature operator, the self-dual and anti-self-dual splitting and the Einstein condition.
• Michael Spivak, A Comprehensive Introduction to Differential Geometry, Volume 2 (Publish or Perish, 1979), for the symmetries of the curvature tensor and the Bianchi identities.