Curvature identities

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1 Introduction

Let M be a smooth manifold and E a vector bundle over M with covariant derivative \nabla. Let R_{ij} = R(\phi_i,\phi_j) = [\nabla_i,\nabla_j] be the curvature tensor with respect to a local parametrization \phi : \R^n_o \to M where \nabla_i = \nabla_{\phi_i} = \nabla_{\partial_i\phi}. By definition,

(1)R_{ij} = -R_{ji}.

If E carries an inner product \langle \,,\,\rangle on each of its fibres and if \nabla is metric, i.e. for any two section s,\tilde s\in \Gamma E the inner product satisfies the product rule

\displaystyle  	\partial_i\langle s,\tilde s\rangle  = \langle \nabla_is,\tilde s\rangle  + \langle s,\nabla_i\tilde s\rangle \,,

then the expression R_{ijkl} = \langle R_{ij}\phi_k,\phi_l\rangle is skew symmetric also in the last two indices:

(2)R_{ijkl} = -R_{ijlk}.

Now suppose that on the tangent bundle TM there is another covariant derivative \nabla which is torsion free, \nabla_i\phi_j = \nabla_j\phi_i. Then the tensor derivative \nabla R of R is defined:

\displaystyle    (\nabla_iR)_{jk} = \nabla_i(R_{jk}) - R(\nabla_i\phi_j,\phi_k) - R(\phi_j,\nabla_i\phi_k).

Abbreviating R(\nabla_i\phi_j,\phi_k) = (ij,k) (symmetric in the first two indices i and j), we have (\nabla_iR)_{jk} = \nabla_iR_{jk} - (ij,k) + (ik,j), using the antisymmetry (1) and the torsion freeness. Now the cyclic sum of the last two

terms vanishes:
\displaystyle -(ij,k)+(ik,j)-(jk,i)+(ji,k) - (ki,j)+(kj,i) = 0.
But the cyclic sum of the first term vanishes also by Jacobi identity, since the tensor derivative of
Tex syntax error
\displaystyle   	\nabla_i(R_{jk}) = [\nabla_i,R_{jk}] = [\nabla_i,[\nabla_j,\nabla_k]].

Thus we obtain an identity for \nabla R, sometimes called the Second Bianchi Identity

(3)(\nabla_iR)_{jk}+(\nabla_jR)_{ki}+(\nabla_kR)_{ij} = 0.

When \nabla is a torsion free covariant derivative on the tangent bundle TM, there is in addition the First Bianchi Identity: Putting R_{ijk} = R_{ij}\phi_k, we have

(4)R_{ijk} + R_{jki} + R_{kij} = 0 .

In fact,

\displaystyle  	\left\{\begin{matrix} & R_{ijk} \cr + & R_{jki} \cr + & R_{kij}\end{matrix}\right\}\ \  	=\ \left\{\begin{matrix} 	&\nabla_i\nabla_j\phi_k &-& \underline{\nabla_j\nabla_i\phi_k} \cr 	 + &\underline{\nabla_j\nabla_k\phi_i} &-& \underline{\underline{\nabla_k\nabla_j\phi_i}} \cr 	 + &\underline{\underline{\nabla_k\nabla_i\phi_j}} &-& \nabla_i\nabla_k\phi_j	\end{matrix}\right\} 	\ \ =\ \ 0

An algebraic consequence is the block symmetry:

(5)R_{ijkl} = R_{klij}

In fact, putting ij|kl := R_{ijkl} and applying (4), (1), (2) we obtain (expressions with equal parentheses cancel each other):

\displaystyle  0 = \left\{\begin{matrix} \ \ \,{ij|kl} \cr +\, (jk|il) \cr +\, [ki|jl] \end{matrix}\right\} +  \left\{\begin{matrix} \ \ \,{ji|lk} \cr +\, \langle il|jk\rangle  \cr + \{lj|ik\} \end{matrix} \right\}- \left\{\begin{matrix} \ \ \,{kl|ij} \cr +\, \langle li|kj\rangle  \cr +\, [ik|lj] \end{matrix} \right\}-  \left\{\begin{matrix} \ \ \,{lk|ji} \cr +\, (kj|li) \cr + \{jl|ki\} \end{matrix}\right\} 	=  2(ij|kl - kl|ij)

Equations (1), (2), (4), (5) are the algebraic identities for the curvature tensor of the Levi-Civita derivative, the Riemannian curvature tensor. By (1), (2), (5), a Riemannian curvature tensor can be viewed as a section of S^2\Lambda^2TM, a symmetric bilinear form on \Lambda^2TM. The antisymmetric 4-forms form another subspace \Lambda^4TM\subset S^2\Lambda^2TM, and the additional identity (4) characterizes precisely the orthogonal complement of \Lambda^4(TM) in S^2\Lambda^2(TM).

2 References

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