SUBMANIFOLDS OF SASAKIAN MANIFOLDS WHICH ARE TANGENT TO THE STRUCTURE VECTOR FIELD (KAEHLERIAN, ANTI-INVARIANT, CURVATURE).

摘要

This dissertation consists of seven sections. Section 0 is a brief outline of basic concepts of Riemannian geometry in general and of the study of Kaehlerian and Sasakian manifolds in particular.;In Section 2 a very broad class of submanifolds of Sasakian manifolds which are tangent to the structure vector field is introduced. These submanifolds are called almost CR submanifolds. There are naturally defined orthogonal distributions D, D* and (mu) on these submanifolds. The fact that these distributions are orthogonal and that they are invariant under a certain operator P allows many of the techniques used in the study of CR submanifolds to be applied. The integrability of some of the various distributions naturally defined on these submanifolds is studied in this section.;In Section 3 parallel and totally geodesic distributions on almost CR submanifolds are studied. This follows the lead of Aurel Bejancu and Bang-Yen Chen who have dealt with similar structures on CR submanifolds of Kaehlerian manifolds.;Attention is focused on Sasakian space forms in Section 4. Formulas for various sectional curvatures are derived. A strong connection between certain sectional curvatures and the operator P is given by Equation 4.4. This is used in Proposition 4.1 to classify all almost CR submanifolds of Sasakian space forms which are of constant curvature.;Section 1 consists of very basic results in the theory of submanifolds of Sasakian manifolds. The most notable result in this section is Proposition 1.2 which implies that a submanifold of a Sasakian manifold is Sasakian if it is invariant under (phi). The definition of a Sasakian submanifold requires tangency to the structure vector field as well as being (phi)-invariant.;The study of submanifolds of Sasakian space forms is continued in Section 5. Various scalar curvatures and mean curvature vectors are defined. Strong relations between some of these sectional curvatures and the eigenvalues of the operator P('2) are discovered. The scalar (rho)(,(xi)) is shown to be 0 if and only if the almost CR submanifold is anti-invariant. The scaler (rho)(,(xi)) is one less than the dimension of the submanifold if and only if the submanifold is invariant. It is an integer if the submanifold is a CR submanifold.;Finally, Section 6 deals with a particular example of an almost CR submanifold.