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

**Sample text**

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.