Theorems of Gauss-Bonnet and Chern-Lashof types in a simply connected symmetric space of non-positive curvature (Q1429099)

The Gauss-Bonnet and Chern-Lashof theorems for an \(n\)-dimensional compact immersed submanifold \(M\) in the \(m\)-dimensional Euclidean space \(\mathbb{R}^m\) involving the Euler characteristic \(\chi(M)\) and the \(k\)th Betti numbers \(b_k(M,F)\), are generalized to compact submanifolds in a simply connected symmetric space \(N\) of non-positive curvature. (It is noted that an inequality of Chern-Lashof type is obtained for the case \(N= H^m(-1)\) by \textit{E. Teufel} in [Colloq. Math. Soc. János Bolyai 46, 1201--1209 (1988; Zbl 0637.53076)]). The proofs are performed by applying the Morse theory to squared distance functions. The corollaries are made for the cases when \(N\) is \(FH^m(c)\), where \(F= \mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) or Cay and \(m= 2\) when \(F= \text{Cay}\). As an example the geodesic sphere in \(FH^m(c)\) is considered.