Differential geometry MOC

Covariant derivative

Let be a -manifold equipped with an affine connexion . The covariant derivative gives a coördiante-independent notion of the directional derivative of any tensor field along a vector field. #m/def/geo/diff

We extend the connexion as a differential operator to tensor fields of arbitrary rank to get an -linear map

which is compatible with

• the exterior derivative on scalar fields so that

for ;

• index contraction so that for ;
• the Leibniz rule so that for and .

Bibliography

• 2009. General relativity, p. 31

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