By Oldrich Kowalski, Emilio E. Musso, Domenico Perrone

This ebook is targeted at the interrelations among the curvature and the geometry of Riemannian manifolds. It comprises learn and survey articles in accordance with the most talks added on the overseas Congress

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**Extra resources for Complex, contact and symmetric manifolds: In honor of L. Vanhecke**

**Example text**

Let’s consider the body Q(τ ), bounded Dτ and H˜ n (τ ) for small τ . At any boundary point there exists a local supporting horosphere. It is a global supporting horosphere too, and the body Q(τ ) is situated in the horoball bounded by supporting horospheres. In other words, the body Q(τ ) is h-convex. Let τ ∗ be a supremum value of τ , for which the body Q(τ ) is h-convex, D∗ = Dτ . Let’s show τ ∗ = ∞. Let us assume the contrary. There are three possible cases: a) D∗ = F n ; b) D∗ = F n and, on the boundary S ∗ of the domain D∗ , there are critical points of the function f = τ/F n .

Question. Is a Riemannian manifold, whose unit tangent sphere bundle is H -contact, a two-point homogeneous space? 4 Semi-symmetric unit tangent sphere bundles ¯ g) A Riemannian manifold (M, ¯ is said to be semi-symmetric if its curvature tensor R¯ satisﬁes: ¯ R(X, Y ) · R¯ = 0 , (12) ¯ ¯ This is equivalent for all vector ﬁelds X, Y , where R(X, Y ) acts as a derivation on R. ¯ the same as the curvature tensor of a symmetric to saying that R¯ p is, for each p ∈ M, space. This last space may vary with p.

T1 M 2 , g) ¯ (equivalently, (T1 M 2 , 2 gS )) is semi-symmetric if and only if (M , g) is either ﬂat or locally isometric to S 2 (1). Proof. If (M 2 , g) has constant Gaussian curvature 0 or 1, from Theorem 3 it follows that T1 M 2 is locally symmetric. In particular, it is semi-symmetric. In order to prove the converse, we need the description of the contact metric structure of T1 M 2 . Using isothermal local coordinates (x 1 , x 2 ) on M 2 , its Riemannian metric g is given by g = e2f ((dx 1 )2 + (dx 2 )2 ), where f is a C ∞ function on M 2 .