The Gauss-Bonnet and Chern-Lashof theorems in a simply connected symmetric space of compact type (Q819516)

Analogues of the theorems of Gauss-Bonnet and Chern-Lashof are proved for an \(n\)-dimensional compact submanifold \(M\) immersed in a simply connected symmetric space \(N\) of compact type: The first type of theorems concerns integral equations between the mean curvatures \(H_i\) and the Euler characteristic of \(M\). The other deals with integral inequalities between the values \(H_i\) and the Betti numbers of \(M\). In particular, the case of \(N\) being an \(m\)-dimensional sphere \(S^m\) is discussed. For instance, in the case \(N = S^m\) and \(M = \gamma\) is a closed curve on \(S^m\) two inequalities for the total curvature of \(\gamma\) are derived. The proofs are based on Morse theory which is applied to squared distance functions.