By Victor Guillemin and Reyer Sjamaar

It is a monograph on convexity houses of second mappings in symplectic geometry. the elemental bring about this topic is the Kirwan convexity theorem, which describes just like a second map when it comes to linear inequalities. This theorem bears a detailed courting to puzzling outdated puzzles from linear algebra, resembling the Horn challenge on sums of Hermitian matrices, on which huge development has been made lately following a step forward by way of Klyachko. The booklet provides an easy neighborhood version for the instant polytope, legitimate within the ""generic"" case, and an straight forward Morse-theoretic argument deriving the Klyachko inequalities and a few in their generalizations. It stories a number of infinite-dimensional manifestations of second convexity, equivalent to the Kostant style theorems for orbits of a loop staff (due to Atiyah and Pressley) or a symplectomorphism staff (due to Bloch, Flaschka and Ratiu). eventually, it provides an account of a brand new convexity theorem for second map photos of orbits of a Borel subgroup of a posh reductive workforce performing on a Kahler manifold, in keeping with potential-theoretic tools in numerous advanced variables. This quantity is usually recommended for self sustaining learn and is appropriate for graduate scholars and researchers attracted to symplectic geometry, algebraic geometry, and geometric combinatorics.

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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.