Metric Compatible Covariant Derivatives

Details Metric Compatible Covariant Derivatives Metric Compatible Covariant Derivatives

2005-01-31

arxiv.org/abs/math/0501561v2

Mathematics

Differential Geometry

Scientific paper

This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in previous papers of the series to introduce through the concept of a metric extensor field g a metric structure for a smooth manifold M. The associated Christoffel operators, a notable decomposition of that object and the associated Levi-Civita connection field are given. The paper introduces also the concept of a geometrical structure for a manifold M as a triple (M,g,gamma), where gamma is a connection extensor field defining a parallelism structure for M . Next, the theory of metric compatible covariant derivatives is given and a relationship between the connection extensor fields and covariant derivatives of two deformed (metric compatible) geometrical structures (M,g,gamma) and (M,eta,gamma') is determined.

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