is called a coherent tangent bundle
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 , 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.