Abstract
A vector field X along a parametrized curve α: I → S in an n-surface S is tangent to S along α if X(t) ∈ S α (t) for all t ∈ I. The derivative Ẋ of such a vector field is. however, generally not tangent to S. We can, nevertheless, obtain a vector field tangent to S by projecting Ẋ(t) orthogonally onto S α (t) for each t ∈ I (see Figure 8.1). This process of differentiating and then projecting onto the tangent space to S defines an operation with the same properties as differentiation, except that now differentiation of vector fields tangent to S yields vector fields tangent to S. This operation is called covariant differentiation.
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© 1979 Springer-Verlag New York Inc.
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Thorpe, J.A. (1979). Parallel Transport. In: Elementary Topics in Differential Geometry. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6153-7_8
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DOI: https://doi.org/10.1007/978-1-4612-6153-7_8
Publisher Name: Springer, New York, NY
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