is called a coherent

tangent bundle over [M.sup.n].

Some substantially revised from the presentations, 22 papers discuss such topics as exact solutions to a Muskat problem with line distributions of sinks and sources, some inequalities for holomorphic self-maps of the unit disc with two fixed points, elliptic perturbations of dynamical systems with a proper node, a porosity theorem for a class of non-expansive set-valued mappings, and a shadow problem for the

tangent bundle of straight lines on a sphere.

Its fibrewise

tangent bundle, which is an m-dimensional real vector bundle over [~.X], will be denoted by [tau](p).

Let us remark that all the previous studies involve only the

tangent bundle TM of M.

It is thanks to the fact that the cohomology class of (2,1)-forms is isomorphic to the cohomology class [H.sup.1/[partial derivative]] (TM), the first Dolbeault cohomology group of M with values in a holomorphic

tangent bundle TM that characterizes infinitesimal complex structure deformations.

In [5], the energy of a unit vector field X on a Riemannian manifold M is defined as the energy of the mapping X : M [right arrow] [T.sup.1] M, where the unit

tangent bundle [T.sup.1] M is equipped with the restriction of the Sasaki metric on TM.

This implies that the

tangent bundle forms an isomorphic group to ([R.sup.1]).

Fano manifolds with nef

tangent bundle and large Picard number Kiwamu WATANABE Communicated by Shigefumi MORI, M.J.A.

We denote the metrics of [bar.M] and its submanifold M by the same letter g, TM is the

tangent bundle of M, and [T.sup.[perpendicular to]] M is the normal bundle of M.

For the plane section [e.sub.i] [conjunction] [e.sub.j] of the

tangent bundle TM spanned by the vectors [e.sub.i] and [e.sub.j] ([e.sub.i] [not equal to] [e.sub.j]) the scalar curvature of M is defined by [kappa] = [[summation].sup.n.sub.i,j=1] K([e.sub.i] [conjunction] [e.sub.j]), where K denotes the sectional curvature of M.

A Finsler manifold M has a

tangent bundle [pi]: TM [right arrow] M.