This is the content of a talk given by the author at the 2009 Lehigh University Geometry/Topology Conference. Using the definition of connection given by Dieudonné, the Sasaki metric on the tangent bundle to a Riemannian manifold is expressed in a natural way. Also, the following property is established. The induced metric on the tangent bundle of an isometrically embedded submanifold is the Sa...

We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural " tangent 2-bundle, " analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relati...

In this paper, we investigate Murre’s conjectures on the structure of rational Chow groups and exhibit explicit Chow–Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle explicit Chow–Künneth projectors are obtained which satisfy Murre’s conjectures and the motiv...

In this paper we explore the idea of looking at the Dirac quantisation conditions as h̄-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle of phasespace, we handle the above constraints using the standard theory of Dirac fo...

We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural " tangent 2-bundle, " analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relati...

The space LV of free loops on a manifold V inherits an action of the circle group T. When V has an almost-complex structure, the tangent bundle of the free loopspace, pulled back to a certain infinite cyclic cover g LV , has an equivariant decomposition as a completion of TV ⊗ (⊕C(k)), where TV is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of TV along evalua...

Finsler geometry motivates a generalization of the Riemannian structure of spacetime to include dependence of the spacetime metric and associated invariant tensor fields on the four-velocity coordinates as well as the spacetime coordinates of the observer. It is then useful to consider the tangent bundle of spacetime with spacetime in the base manifold and four-velocity space in the fiber. A ph...

We write down the local equations that characterize the sub-manifolds N of a Dirac manifold M which have a normal bundle that is either a coisotropic or an isotropic submanifold of T M endowed with the tangent Dirac structure. In the Poisson case, these formulas prove again a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of T M with the tangent Poisson s...

Let M be a close complex manifold and T M its holomorphic tangent bundle. We prove that if the global holomorphic sections of tangent bundle generate each fibre, then M is a complex homogeneous manifold. It implies that every irreducible close Kähler manifold with ample tangent bundle is isomorphic to the projective space. This provides an alternative proof of Hartshore's conjecture in algebrai...

In this paper we study a Riemanian metric on the tangent bundle T (M) of a Riemannian manifold M which generalizes the Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T (M) a structure of locally conformal almost Kählerian manifold. We found conditions under which T (M) is almost Kählerian, locally conformal Kählerian or Kählerian or wh...