In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation (M,F) with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O’Neill tensor and the first eigenvalue of the Dirac operator on M . We discuss examples and also define a new version of the basic Laplacian wh...

The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric (X,X) ≥ a (n − 1) |X| for every X ∈ TM for some fixed a > 0, then the smallest positive eigenvalue λ of the Laplacian satisfies λ ≥ an. The Obata theorem states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvatu...

In this note, we discuss Riemannian foliations, which are smooth foliations that have a transverse geometric structure. We explain a known generalization of Dirac-type operators to transverse operators called “basic Dirac operators” on Riemannian foliations, which require the additional structure of what is called a bundle-like metric. We explain the result in [10] that the spectrum of such an ...

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M , about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable sp...

Let S be a compact Riemann surface (holomorphic curve) of genus g. Let p1, p2, · · · , ps be s > 0 points on it; these points define a divisor, and we denote the open Riemann surface S \ {p1, . . . , ps} by S. When 3g− 3+ s > 0, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points p1, p2, · · · , ps then become cusps at infinity. Even in the remain...

The main purpose of this paper is to introduce several concepts of the metric-like spaces. For instance, we define concepts such as equal-like points, cluster points and completely separate points. Furthermore, this paper is an attempt to present compatibility definitions for the distance between a point and a subset of a metric-like space and also for the distance between two subsets of a metr...

This paper addresses Cheeger and Gromoll’s question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space which together satisfy a ...

in this paper,  the concept of fuzzy metric-like spaces is introduced which generalizes  the notion of fuzzy metric spaces given by george and veeramani cite{vee1}. some fixed point results for fuzzy contractive mappings on fuzzy metric-like spaces are derived. these results generalize several comparable results from the current literature. we also provide illustrative examples in support of ou...

This article is devoted to a study of flat orbifold vector bundles. We construct bijection between the isomorphic classes proper bundles and equivalence representations fundamental groups base orbifolds. establish Bismut-Zhang like anomaly formula for Ray-Singer metric on determine line cohomology compact with coefficients in an bundle. show that analytic torsion acyclic unitary bundle equal va...

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal, Whitney sum E⊕C where E is a given Courant algebroid and C is a flat, pseudo-Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, that is, a Courant algebroid with a surjective anchor, and describ...