In this paper we will use our results about the local deformation theory of Calabi-Yau manifolds to define a Higgs field on the tangent bundle of the moduli space of CY threefolds. Combining this Higgs field with the Levi-Chevitta connection of the Weil-Petersson metrics on the moduli space of three dimensional CY manifolds, we construct a new Sp(2h,R) connection, following the ideas of Cecotti...

We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable. Also, we find geometric properties of both leaves of Liouville distribution and the vertical distribution.

A neural network taken as a model of a trainable system appears to be nothing but a dynamical system evolving on a tangent bundle with changeable metrics. In other words to learn means to change metrics of a definite manifold.

We prove that the indices of fibered-cusp and d-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula for the Dirac operator of horn-cone and of fibered horn metrics. Let X be a comp...

We prove that the indices of fibered-cusp and d-Dirac operators on a spin manifold with fibered boundary coincide if the associated family of Dirac operators on the fibers of the boundary is invertible. This answers a question raised by Piazza. Under this invertibility assumption, our method yields an index formula for the Dirac operator of horn-cone and of fibered horn metrics. Let X be a comp...

We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for C1+ε flows on closed smooth three dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed C∞ surface has at least const×(ehT /T ) simple closed orbits of period less than T , whenever the topological entropy h is positive – and witho...

For a brief history of the problem and a proof under stringent additional conditions see [2]. Let us first recall some of the known facts about such surfaces which we shall use. Proofs may be found in [l, §102]. Normalize the conjugate distance so it equals T. Then every geodesic is closed and has length 2ir. The mapping of any point into its conjugate point (determined uniquely, independently ...

The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations. MSC 2000: 70G45, 70H03, 70H05

For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of HeinonenKoskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.

In this paper, we obtain a Nakano type inequality for vertical valued ∂h harmonic mixed forms with compact support on the total space of the holomorphic tangent bundle of a complex Finsler manifold. AMS subject classifications: 53B40, 32L20