Showing papers on "Distribution (differential geometry) published in 1981"

Differential geometry of real submanifolds in a Kähler manifold

Abstract: LetN be a real submanifold in a complex manifoldM. If the maximal complex subspaces of the tangent spaces ofM contained in the tangent spaces ofN are of constant dimension and they define a differentiable distribution, thenN is called a generic submanifold. The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds andCR-submanifolds. In this paper we initiate a study of generic submanifolds in a Kahler manifold from differential geometric point of view. Some fundamental results in this respect will be obtained.

Quaternion CR-submanifolds of quaternion manifolds

Abstract: A quaternion manifold (or quaternion Kaehlerian manifold [10]) is defined as a Riemannian manifold whose holonomy group is a subgroup of Sp(l). Sp(m)=Sp(l)xSp(m)/{±1}. The quaternion projective space QP, its noncompact dual and the quaternion number space Q are three important examples of quaternion manifolds. It is well-known that on a quaternion manifold M, there exists a 3-dimensional vector bundle E of tensors of type (1, 1) with local cross-section of almost Hermitian structures satisfying certain conditions (see § 2 for details). A submanifold N in a quaternion manifold M is called a quaternion (respectively, totally real) submanifold if each tangent space of N is carried into itself (respectively, the normal space) by each section in E. It is known that every quaternion submanifold in any quaternion manifold is always totally geodesic. So it is more interesting to study a more general class of submanifolds than quaternion submanifolds. The main purpose of this paper is to establish the general theory of quaternion Ci?-submanifolds in a quaternion manifold which generalizes the theory of quaternion submanifolds and the theory of totally real submanifolds. It is proved in section 3 that such submanifolds are characterized by a simple equation in terms of the curvature tensor of a quaternion-space-form. In section 4 we shall study the integrability of the two natural distributions on a quaternion Ci?-submanifold. In section 5 we obtain some basic lemmas for quaternion Ci?-submamfolds. In particular, we shall obtain two fundamental lemmas which play important role in this theory. Several applications of the fundamental lemmas are given in section 6. In section 7 we study quaternion C7?-submanifolds which are foliated by totally geodesic, totally real submanifolds. In the last section we give an example of a quaternion Ci?-submanifold of an almost quaternion metric manifold on which the totally real distribution is not integrable.

Apparatus for delivering fluid to a rotating body

Abstract: An apparatus is disclosed for feeding a fluid to a rotary body such as a cylindrical rotary mill. A rotary annular manifold is connected to the mill. A plurality of orifices in the face of the manifold, each with normally closed valves, allows the fluid to enter the manifold for subsequent distribution to the rotary mill. Fluid is transmitted to the rotary manifold from a stationary manifold. Fluid is transmitted to the stationary manifold from a fluid source. The stationary manifold has a cavity which is positioned against the rotary manifold and aligned over several of the orifices. As fluid, under pressure, enters the cavity in the stationary manifold, the fluid pressure opens the orifice valves and enters the rotary manifold. As the manifold rotates, orifices moving past the cavity close preventing the escape of fluid and subsequent orifices pass in front of the cavity allowing more fluid to enter the rotary manifold. The stationary manifold is held against the rotary manifold with a pressure at least equal to the parting force of the fluid in the cavity of the stationary manifold.