Curvature identities
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1 Introduction
Let be a smooth manifold and a vector bundle over with covariant derivative . Let be the curvature tensor with respect to a local parametrization where . By definition, \beq R_{ij} = -R_{ji}. \eeq If carries an inner product on each of its fibres and if is {\it metric}, i.e.\ for any two section the inner product satisfies the product rule
then the expression is skew symmetric also in the last two indices:
Now suppose that on the tangent bundle there is another covariant derivative which is torsion free, . Then the tensor derivative of is defined:
Abbreviating (symmetric in the first two indices and ), we have , using the antisymmetry (1) and the torsion freeness. Now the cyclic sum of the last two
terms vanishes:Tex syntax erroris
Thus we obtain an identity for , sometimes called {\it Second Bianchi Identity}
When is a torsion free covariant derivative on the tangent bundle , there is in addition the {\it First Bianchi Identity}: Putting , we have
In fact,
An algebraic consequence is the block symmetry:
In fact, putting and applying (3), (1), (1) we obtain (expressions with equal parentheses cancel each other):
Equations (1), (1), (3), (4) are the algebraic identities for the curvature tensor of the Levi-Civita derivative, the {\it Riemannian curvature tensor}. By (1), (1), (4), a Riemannian curvature tensor can be viewed as a section of , a symmetric bilinear form on . The antisymmetric 4-forms form another subspace , and the additional identity (3) characterizes precisely the orthogonal complement of in .