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,
If carries an inner product on each of its fibres and if is 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 the Second Bianchi Identity
When is a torsion free covariant derivative on the tangent bundle , there is in addition the First Bianchi Identity: Putting , we have
In fact,
An algebraic consequence is the block symmetry:
In fact, putting and applying (4), (1), (2) we obtain (expressions with equal parentheses cancel each other):
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 , a symmetric bilinear form on . The antisymmetric 4-forms form another subspace , and the additional identity (4) characterizes precisely the orthogonal complement of in .