Geometry of submanifolds in Euclidean spaces (Q2781470)

The notes under review are written for a course held at the University of Coimbra in Portugal, September 4--7, 2001. They cover some basic notions for the study of submanifolds of arbitrary codimension in Euclidean spaces. The lectures center around the notion of a focal point.NEWLINENEWLINEThe notion of isoparametric submanifolds is introduced. A submanifold \(M\) of \(\mathbb R^n\) is said to be isoparametric if its normal bundle is flat and if the eigenvalues of the Weingarten operators \(A_{\xi}\) for every normal vector field \(\xi\) are constant.NEWLINENEWLINEThe concepts are illustrated through the following examples of submanifolds of \(\mathbb R^n\): the orthogonal group \(O(n)\) and the isospectral submanifolds of symmetric matrices.