Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree

For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.

     

Conformal vector fields on tangent bundle of a Riemannian manifold

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM,G).     

Some cross-section theorems on the tangent bundle over a finslerian manifold

Summary Let T(M) be the tangent bundle over a Finslerian manifold M of n-dimension endowed with the Cartan connection ∇. One makes T(M) into a 2n dimensional affinely connected manifold by assigning a connection ∇^c to T(M). The cross-section of a vector field V defined in M reveals in T(M) an n-dimensional submanifold and its geometry is developed by means of the affine subspace theory and of the affine collineations in the base Finsler manifold. This work was supported by the National Research Council of Canada, 1970–1971, A-4037. Entrata in Redazione il giorno 8 maggio 1971.     

On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, ifM is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection ofM is quasi-Einstein, too, provided thatM is tangent to the Lee field ofM. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ _m (of a complex Hopf manifoldS ^2m+1 ×S ¹).     

Estimation for jump processes in the tangent bundle of a Riemann manifold

A stochastic process is formulated in the tangent bundle of a Riemann manifold where the vector fibre portion of the process is a jump process. Since the tangent spaces change as the process in the base manifold evolves, it is necessary to define a jump process in the fibres of the tangent bundle with respect to the process in the base manifold. An estimation problem is formulated and solved for a process obtained from the jump process in the fibres of the tangent bundle where the observations include the process in the base manifold and the jump times. Since each fibre of the tangent bundle is a linear space, a suitable modification of some results for estimation in linear spaces can be used to solve the aforementioned estimation problem.Research supported by NSF Grants ENG 75-06562 and MCS 76-01695 and AFOSR Grant 77-3177.     

On the volume of the projectivized tangent bundle in a complex Finsler manifold

We prove that the volume of PT _z ^{1, 0} M, calculated with respect to a Kähler metric induced by a complex Finsler structure, is a constant. This contrasts sharply with the situation in real Finsler geometry, where the volume of unit tangent sphere at each point x in a real Finsler manifold is in general a function of x. Furthermore, we point out that different metrics have different constants in general.     

Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with ${\nabla=D + S}In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with induced by the one-parameter family of connections given by and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-K?hler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt ^*-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds (M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt ^*-bundles generalised para-pluriharmonic maps into , respectively, into SO ₀(n,n)/U ^π(C ⁿ ), where U ^π(C ⁿ ) is the para-complex analogue of the unitary group.      

Vortex equation in holomorphic line bundle over complex manifold

In this note we study the vortex equation in holomorphic line bundle over non-Kähler complex manifolds. We prove a existence theorem to that equation by means of the upper and lower solution method to some Kazdan-Warner type equation.     

The fibre bundle of degenerate tangent planes of a Lorentzian manifold and the smoothness of the null sectional curvature

The Grassmann bundle of degenerate tangent planes to a Lorentzian manifold is introduced and several of its properties are deduced. After a suitable normalization, the null sectional curvature associated to a Lorentzian metric may be considered as a well-defined function on the Grassmann bundle of degenerate tangent planes. The smoothness of this function is proved and some consequences are obtained.     

Lightlike real hypersurfaces of indefinite quaternion Kaehler manifold

In this paper, we introduce real lightlike hypersurfaces of indefinite quaternion Kaehler manifold. Fundamental properties of real lightlike hypersurfaces of an indefinite quaternion Kaehler manifold are investigated. We prove the non existence of real lightlike hypersurfaces in indefinite qaternionic space form under some conditions. Received 31 October 2000; revised 20 June 2001.     

Comparison theorems for the volume of a complex submanifold of a Kaehler manifold

LetM be a Kaehler manifold of real dimension 2n with holomorphic sectional curvatureK _H≥4λ and antiholomorphic Ricci curvatureρ _A≥(2n−2)λ, andP is a complex hypersurface. We give a bound for the quotient (volume ofP)/(volume ofM) and prove that this bound is attained if and only ifP=C P ⁿ⁻¹(λ) andM=C P ⁿ(λ). Moreover, we give some results on the volume of of tubes aboutP inM. Work partially supported by a DGICYT Grant No. PS87-0115-CO3-01.