The Kähler-Einstein metrics on the fibers of an effectively parameterized family of canonically polarized manifolds induce a hermitian metric on the relative canonical bundle. We use a global elliptic equation to show that this metric is strictly positive. Applications concern the curvature of the classical and generalized Weil-Petersson metrics and hyperbolicity of moduli spaces.

We compute the relation between the Quillen metric and and the canonical metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we present a general formula relating the ζdeterminant and the canonical determinant for a class of higher-order elliptic boundary value problems.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n ≥ 3. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat.

for the first time in the minkowski space, an rn 1 time-like complementary ruled surface isdescribed and relations are given connected with an asymptotic and tangential bundle of the time-likecomplementary ruled surface. furthermore, theorems are given related to edge space, central space andcentral ruled surface of this complementary ruled surface.

and Applied Analysis 3 in the normal bundle T⊥M, and AN is the shape operator of the second fundamental form. Moreover, we have g ANX, Y g h X,Y ,N , 2.4 where g denotes the Riemannian metric onM as well as the metric induced onM. The mean curvature vector H on M is given by

We present a simple derivation of the Ricci-flat Kähler metric and its Kähler potential on the canonical line bundle over arbitrary Kähler coset space equipped with the Kähler-Einstein metric. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]

Let Fm = (M,F ) be a Finsler manifold and G be the Sasaki– Finsler metric on the slit tangent bundle TM0 = TM {0} of M . We express the scalar curvature ρ̃ of the Riemannian manifold (TM0, G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ρ̃ to be a positively homogenenous function of degree zero with respect to the fiber coo...

1 1 INTRODUCTION 2 1 Introduction Classical calculus is a basic tool in analysis. We use it so often that we forget that its construction needed considerable time and effort. Especially in the last decade, the progresses made in the field of analysis in metric spaces make us reconsider this calculus. Along this line of thought, all started with the definition of Pansu derivative [24] and its ve...

When a Hamiltonian system has a \Kinetic + Potential" structure, the resulting ow is locally a geodesic ow. But there may be singularities of the geodesic structure, so the local structure does not always imply that the ow is globally a geodesic ow. In order for a ow to be a geodesic ow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions ...