Riemannian Submanifolds

  • John M. LeeEmail author
Part of the Graduate Texts in Mathematics book series (GTM, volume 176)


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.

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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