We consider a connected symplectic manifold M acted on by a connected Lie group G in a Hamiltonian fashion. If G is compact we study the smooth function f =‖ μ ‖. We prove that if a point x ∈ M realizes a local maximum of the squared moment map ‖ μ ‖ then the orbit Gx is symplectic and Gμ(μ(x)) is G-equivariantly symplectomorphic to a product of a flag manifold and a symplectic manifold which i...

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and con...

Put in another way, for a compact manifold with boundary, if we know that the boundary is S(intrinsic geometry on the boundary) and totally geodesic (extrinsic geometry) then we recognize the manifold as the hemisphere S+, provided Ric ≥ (n− 1) g. To put this result in a context, we first recall the following Theorem 2. Let (M, g) be a compact Riemannian manifold with boundary and scalar curvat...

Given a compact Riemannian manifold on which a compact Lie group acts by isometries, it is shown that there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. Furthermore, this isometry (and foliation) may be chosen so that a leaf closure is mapped to an orbit with the same volume, even though the dimension o...

A stationary space-time (M,g) is a 4-manifold M with a smooth Lorentzian metric g, of signature (−,+,+,+), which has a smooth 1-parameter group G ≈ R of isometries whose orbits are time-like curves in M . We assume throughout the paper that M is a chronological space-time, i.e. M admits no closed time-like curves, c.f. §1.1 for further discussion. Let S be the orbit space of the action G. Then ...

Theorem 1. Suppose (M; g) is a noncompact complete Riemannian manifold with Ric (n 1), we have 0 (n 1) =4. The estimate is sharp since the spectrum is the ray [(n 1) =4;+1) for the hyperbolic space H. For Kähler manifolds, this estimate can be improved. On a Kähler manifold (M; g) of complex dimension n, where g is the Riemannian metric, let ! = g (J ; ) be the Kähler form. In local holomorphic...

Klein defined geometry in terms of invariance under groups actions; here we give a discrete (partial) converse of this, interpreting all (finitely presentable) groups in terms of the geometry of hyperbolic 3-manifolds (whose fundamental groups are, appropriately, Kleinian groups). For G∗ a Kleinian group of isometries of hyperbolic 3-space H, with MG∗ ∼= H3/G∗ a non-compact N -cusped orientable...

In this paper we study the question of the existence of a continuous inverse to the multiplication mapping (/, g) -» (fg, g) defined on pairs of C°° functions on a manifold M. Obviously, restrictions must be imposed on the domain of such an inverse. This leads us to the study of a modified problem: Find an appropriate domain for the inverse of (/, G) -» (/(/> ° G), G), where G is a C°° mapping ...

We prove that if (M, g) is a compact locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature (ii) (M, g) is isometric to a locally symmetric space (iii) (M, g) is Kähler and biholomorphic to CP n. This is implied by the following two results: (i) Let (M, g) be a compact, l...