By Oldrich Kowalski, Emilio E. Musso, Domenico Perrone

ISBN-10: 0817638504

ISBN-13: 9780817638504

This ebook is concentrated at the interrelations among the curvature and the geometry of Riemannian manifolds. It includes learn and survey articles in line with the most talks introduced on the foreign Congress

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**Sample text**

M, ¯ is H -contact if and only if ξ is an eigenvector for the Ricci operator. K-contact spaces (in particular, Sasakian manifolds), (k, µ)-spaces, locally ϕsymmetric spaces are all examples of H -contact manifolds. We can refer to [P5] for more details on H -contact spaces. As concerns the unit tangent sphere bundle, Theorem 1 of [BV3] can be reformulated in the following way: ¯ is H Theorem 14 ([BV3]). If (M, g) is two-point homogeneous, then (T1 M, η, g) contact. Note that, according to Theorems 2, 12 and 14, if (M, g) is a two-point homogeneous space of non-constant sectional curvature, then its unit tangent sphere bundle ¯ is an H -contact space which is neither locally ϕ-symmetric (in particular, (T1 M, η, g) it is not a (k, µ)-space), nor K-contact.

Note that the value k coth kr is the geodesic curvature of a circumference of radius r in the Lobachevsky plane of curvature −k 2 . An orientable regular (C 2 or more) hypersurface F of a Hadamard manifold M is λ-convex if, for a selection of its unit normal vector, the normal curvature kn of F satisﬁes kn ≥ λ. A domain ⊂ M is λ-convex if for every point P ∈ ∂ there is a regular λ-convex hypersurface F through P leaving a neighborhood of P in the convex side (the side where the unit normal vectors points) of F .

It is either nonnegative or nonpositive everywhere); (ii) ϕ(M) is a convex hypersurface. By Gaussian curvature, we mean the product of the principal curvatures. S. Alexander generalized the Hadamard theorem for compact hypersurfaces in any complete, simply connected Riemannian manifold of nonpositive sectional curvature [3]. A topological immersion f : N n → M of a manifold N n into a Riemannian manifold M is called locally convex at a point x ∈ N n if x has a neighborhood U such that f (U ) is a part of the boundary of a convex set in M.

### Complex, contact and symmetric manifolds: In honor of L. Vanhecke by Oldrich Kowalski, Emilio E. Musso, Domenico Perrone

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