Submanifold Theory

This book provides a comprehensive introduction to Submanifold theory, focusing on general properties of isometric and conformal immersions of Riemannian manifolds into space forms. One main theme is the isometric and conformal deformation problem for submanifolds of arbitrary dimension and codimension. Several relevant classes of submanifolds are also discussed, including constant curvature submanifolds, submanifolds of nonpositive extrinsic curvature, conformally flat submanifolds and real Kaehler submanifolds. This is the first textbook to treat a substantial proportion of the material presented here. The first chapters are suitable for an introductory course on Submanifold theory for students with a basic background on Riemannian geometry. The remaining chapters could be used in a more advanced course by students aiming at initiating research on the subject, and are also intended to serve as a reference for specialists in the field.

The basic equations of a submanifold.- Reduction of codimension.- Minimal submanifolds.- Local rigidity of submanifolds.- Constant curvature submanifolds.- Submanifolds with nonpositive extrinsic curvature.- Submanifolds with relative nullity.- Isometric immersions of Riemannian products.- Conformal immersions.- Isometric immersions of warped products.- The Sbrana-Cartan hypersurfaces.- Genuine deformations.- Deformations of complete submanifolds.- Innitesimal bendings.- Real Kaehler submanifolds.- Conformally at submanifolds.- Conformally deformable hypersurfaces.- Vector bundles.