Two theorems on Kaehler manifolds.

Lecture on Kaehler manifolds

Ricci tensor for $GCR$-lightlike submanifolds of indefinite Kaehler manifolds

We obtain the expression of Ricci tensor for a $GCR$-lightlikesubmanifold of indefinite complex space form and discuss itsproperties on a totally geodesic $GCR$-lightlike submanifold of anindefinite complex space form. Moreover, we have proved that everyproper totally umbilical $GCR$-lightlike submanifold of anindefinite Kaehler manifold is a totally geodesic $GCR$-lightlikesubmanifold.

Vanishing theorems on Hermitian manifolds

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddc-harmonic Kähler form and positive (1, 1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a coroll...

Vanishing Theorems on Covering Manifolds

Let M be an oriented even-dimensional Riemannian manifold on which a discrete group Γ of orientation-preserving isometries acts freely, so that the quotientX = M/Γ is compact. We prove a vanishing theorem for a half-kernel of a Γ-invariant Dirac operator on a Γ-equivariant Clifford module overM , twisted by a sufficiently large power of a Γ-equivariant line bundle, whose curvature is non-degene...

Mean Value Theorems on Manifolds

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as the results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to ‘heat spheres’ is proved for heat equation with respect to evolving Riemannian metrics via a space-time consideration. Som...