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

By Oldrich Kowalski, Emilio E. Musso, Domenico Perrone

This ebook is concentrated at the interrelations among the curvature and the geometry of Riemannian manifolds. It comprises study and survey articles according to the most talks brought on the overseas Congress

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We conclude that Xn belongs to the nullity distribution of the curvature tensor Rx . In this case, the conditions (12), (14) and (16) are trivially satisfied; — if n is odd, the eigenvalue −4 has odd multiplicity n − 2 on {Xj , Xn }⊥ . The eigenvalue corresponding to Xn must then be −4 as well. So, it holds, |R(u, Yj )Xn |2 = 4, for j = 1, . . , n − 1. It even holds, |R(u, Y )Xn |2 = 4, for every unit vector Y orthogonal to u and g(R(u, Y )Xn , R(u, Z)Xn ) = 4g(Y, Z), for all vectors Y and Z orthogonal to u.

T1 M, η, g) ¯ is locally ϕ-symmetric if and only if (M, g) has constant sectional curvature. Contact Metric Geometry of the Unit Tangent Sphere Bundle 49 H -contact unit tangent sphere bundles A unit vector field V on a Riemannian manifold (M, g) determines a map between (M, g) itself and its unit tangent sphere bundle (T1 M, gS ). If M n is compact and orientable, the energy of V is defined as the energy of the corresponding map: E(V ) = 1 2 dV M 2 dv = 1 n vol(M, g) + 2 2 ∇V 2 dv. M V is called harmonic if it is a critical point for E in the set of all unit vector fields of M [Wo].

The author, in joint works with E. Boeckx and D. Perrone, obtained the following generalization of Theorem 3 by Blair: Theorem 16 ([BC], [CP2]). If the unit tangent sphere bundle (T1 M, gS ) (equivalently, (T1 M, η, g)) ¯ of a Riemannian manifold (M, g) is semi-symmetric, then it is locally symmetric. Therefore, (T1 M, gS ) is semi-symmetric if and only if either (M, g) is flat or it is locally isometric to S 2 (1). In order to prove this result, after recalling the local structure of a semi-symmetric space, we deal separately with the cases when T1 M is three-dimensional, where we make use of the special features of a three-dimensional contact metric manifold, locally irreducible and locally reducible.

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