We calculate the Riemann curvature tensor and sectional curvature for the Lie group of volume-preserving diffeomorphisms of the Klein bottle and projective plane. In particular, we investigate the sign of the sectional curvature, and find a possible disagreement with a theorem of Lukatskii. We suggest an amendment to this theorem.

(1.1) Let c be a real number. Represent by A4"(c) a n-dimensional space form of curvature c. Let M N be a N-dimensional connected Riemannian manifold. The question that served as the starting point for this paper was to find simple conditions on the metric of M so that, if f:M"~ffl"+P(c), p>= 1, is an isometric minimal immersion then f is rigid in the following sense: given another minimal imme...

The classical Cartan-Hadamard theorem asserts that a closed Riemannian manifold M with non-positive sectional curvature has universal cover M̃ diffeomorphic to R, and a byproduct of the proof is that ∂∞M̃n is homeomorphic to Sn−1. We prove analogues of these two results in the case where M has a non-empty totally geodesic boundary. More precisely, if M 1 ,M n 2 are two negatively curved Riemannia...