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Riemannian Submanifolds

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Part of the Graduate Texts in Mathematics book series (GTM,volume 176)

Abstract

This chapter has a dual purpose: first to apply the theory of curvature to Riemannian submanifolds, and then to use these concepts to derive a precise quantitative interpretation of the curvature tensor. We first define a vector-valued bilinear form called the second fundamental form, which measures the way a submanifold curves within the ambient manifold. This leads to a quantitative geometric interpretation of the curvature tensor, as an object that encodes the sectional curvatures, which are Gaussian curvatures of 2-dimensional submanifolds swept out by geodesics tangent to 2-planes in a tangent space.

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  • DOI: 10.1007/978-3-319-91755-9_8
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Correspondence to John M. Lee .

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Lee, J.M. (2018). Riemannian Submanifolds. In: Introduction to Riemannian Manifolds. Graduate Texts in Mathematics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-91755-9_8

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