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The Geometry of Physics by Theodore Frankel

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CHAPTER 9

Covariant Differentiation and Curvature

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WE saw in Section 2.4 that the partial derivatives jvi of a vector field v do not form the components of a tensor. For a covariant vector field image1 we did show that we can construct a tensor by taking a combination of partial derivatives, jak − ∂kaj, the exterior derivative, but that jak by themselves do not yield a tensor. Our goal in this chapter is to introduce an added structure to the notion of a manifold, a structure that will allow us to form a generalized derivative, a “covariant” derivative, ...

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