We define transgressions of arbitrary order, with respect to families unit-vector fields indexed by a polytope, for the Pfaffian metric connections semi-Riemannian metrics on vector bundles. apply this formula compute Euler characteristic Riemannian polyhedral manifold, very much in spirit Chern’s differential-geometric proof generalized Gauss–Bonnet closed manifolds and manifolds-with-boundary...

We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

A semi-Riemannian manifold is said to satisfy R ≥ K (or R ≤ K) if spacelike sectional curvatures are ≥ K and timelike ones are ≤ K (or the reverse). Such spaces are abundant, as warped product constructions show; they include, in particular, big bang Robertson-Walker spaces. By stability, there are many non-warped product examples. We prove the equivalence of this type of curvature bound with l...

Formal Riemannian metrics are characterized by the property that all products of harmonic forms are again harmonic. They have been studied over the last ten years and there are still many interesting open conjectures related to geometric formality. The existence of a formal metric implies Sullivan’s formality of the manifold, and hence formal metrics can exist only in presence of a very restric...

We consider the problem of learning a Riemannian metric associated with a given differentiable manifold and a set of points. Our approach to the problem involves choosing a metric from a parametric family that is based on maximizing the inverse volume of a given dataset of points. From a statistical perspective, it is related to maximum likelihood under a model that assigns probabilities invers...

We give a condition on Riemannian submersions from a Riemannian manifold M to a Riemannian manifold N which will ensure that it induces a differential operator on N from the Laplace-Beltrami operator on M . Equivalently, this condition ensures that a Riemannian submersion maps Brownian motion to a diffusion. 2000 Mathematics Subject Classification. Primary 58J65, 53C21.