Introduction to Riemannian Manifolds pp 225-262 | Cite as

# Riemannian Submanifolds

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## 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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