Tangentially Positive Isometric Actions and Conjugate Points
نویسنده
چکیده
Let (M, g) be a complete Riemannian manifold with no conjugate points and f : (M, g) → (B, gB) a principal G-bundle, where G is a Lie group acting by isometries and B the smooth quotient with gB the Riemannian submersion metric. We obtain a characterization of conjugate point-free quotients (B, gB) in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle TM, from which we then derive necessary conditions, involving G and M, for the quotient metric to be conjugate point-free, particularly for M a reducible Riemannian manifold. Let μG : TM → G∗, with G the Lie Algebra of G, be the moment map of the tangential G-action on TM and let GP be the canonical pseudo-Riemannian metric on TM defined by the symplectic form dΘ and the map F : TM → M × M, F (z) = (exp(−z), exp(z)). First we prove a theorem, stating that if GP is not positive definite on the action vector fields for the tangential action along μG −1(0) then (B, gB) acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize self-parallel vector fields on M in terms of the positivity of the GP-length of their tangential lifts along certain canonical subsets of TM. We use this to derive some necessary conditions, on G and M, for actions to be tangentially positive on relevant subsets of TM, which we then apply to isometric actions on complete conjugate point-free reducible Riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions.
منابع مشابه
First Cohomology and Local Rigidity of Group Actions
Given a topological group G, and a finitely generated group Γ, a homomorphism π : Γ→G is locally rigid if any nearby by homomorphism π′ is conjugate to π by a small element of G. In 1964, Weil gave a criterion for local rigidity of a homomorphism from a finitely generated group Γ to a finite dimensional Lie group G in terms of cohomology of Γ with coefficients in the Lie algebra of G. Here we g...
متن کاملStarlike Functions of order α With Respect To 2(j,k)-Symmetric Conjugate Points
In this paper, we introduced and investigated starlike and convex functions of order α with respect to 2(j,k)-symmetric conjugate points and coefficient inequality for function belonging to these classes are provided . Also we obtain some convolution condition for functions belonging to this class.
متن کاملBoundary and Lens Rigidity of Lorentzian Surfaces
LARS ANDERSSON, MATTIAS DAHL, AND RALPH HOWARD Abstract. Let g be a Lorentzian metric on the plane R2 that agrees with the standard metric g0 = −dx + dy outside a compact set and so that there are no conjugate points along any time-like geodesic of (R2 , g). Then (R2 , g) and (R2 , g0) are isometric. Further, if (M, g) and (M∗, g∗) are two dimensional compact time oriented Lorentzian manifolds ...
متن کاملSome Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points
متن کامل
Two-orbit Manifolds with Boundary
In this article we are interested in the classification of some Lie group actions on differentiable manifolds. In full generality, classifying all actions of Lie groups is of course an unachievable task. Let us consider the case when there are very few orbits. The case of transitive actions is easily dealt with: the manifold is then a homogeneous space, and it is sufficient to give the stabiliz...
متن کامل