Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces ...

N. V. Efimov [Efi64] proved that there is no complete, smooth surface in R with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if M has sectional curvature between two constants K2 and K3, then there exists K1 < min(K2, 0) such that M contains no smooth, complete immersed surface with curvature below K1. Optimal values of K1 are dete...

If π : M → B is a Riemannian Submersion and M has positive sectional curvature, O’Neill’s Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from...

Global geometric properties of product manifolds M = M × R2, endowed with a metric type 〈·, ·〉 = 〈·, ·〉R + 2dudv + H(x, u)du 2 (where 〈·, ·〉R is a Riemannian metric on M and H : M × R → R a function), which generalize classical plane waves, are revisited. Our study covers causality (causal ladder, inexistence of horizons), geodesic completeness, geodesic connectedness and existence of conjugate...

We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Introduction Consider a distribution on a manifold M, i.e. subbundle of the tangent bundle Π ⊂ TM . Non-holonomic Riemannian metric is a Riemannian metric g ∈ SΠ on this bundle. We call the pair (Π, g) sub-Riemannian structure. A curve γ : [0, 1] → M is called horizontal if γ̇ is a sect...

In Kirchheim-Magnani [7] the authors construct a left invariant distance ρ on the Heisenberg group such that the identity map id is 1-Lipschitz but it is not metrically differentiable anywhere. In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability. In fact we show that they construct something which fails shortly from being a dilatation...

For a large class of semiclassical pseudodifferential operators, including Schrödinger operators, P (h) = −h∆g + V (x), on compact Riemannian manifolds, we give logarithmic lower bounds on the mass of eigenfunctions outside neighbourhoods of generic closed hyperbolic orbits. More precisely we show that if A is a pseudodifferential operator which is microlocally equal to the identity near the hy...

One of the basic questions of Riemannian geometry is that "If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?". Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), w...

An L theory of differential forms is proposed for the Banach manifold of continuous paths on Riemannian manifolds M furnished with its Brownian motion measure. Differentiation must be restricted to certain Hilbert space directions, the H-tangent vectors. To obtain a closed exterior differential operator the relevant spaces of differential forms, the H-forms, are perturbed by the curvature of M ...