Totally real submanifolds of the nearly kaehler 6-sphere

Abstract

Totally real 3-dimensiunal submanifolds of the nearly Kaehler 6-sphere are the main topic of this thesis. Having introduced preliminaries on the theory of complex and almost complex manifolds, the nearly Kaehler structure of S(^6) and the non existence of almost complex. 4-dimensional submanifolds of the 6-sphere [G3], the results of N. Ejiri [Ejl] on orientability, minimality and characterization by means of constant sectional curvature are given. Results concerning the pinching of the sectional curvature in the compact case are coming next (see: [D.O.V.V.1]. [D.V.V2]). These results are obtained by using the integral formulae of .A. Ros [R]. formulae which play a crucial role in global Riemannian geometry. .After a discussion on a new Riemannian invariant δ. introduced by B.Y.Chen in [Ch2]. For submanifolds of real space forms and the inequality (which is the best possible) satisfied by S, we focus on the case where the inequality becomes an equality. In this case the shape operator of the immersion attains a special form and this helps with the classification. In particular, if M is a 3-dimensional totally real submanifold of S(^6) then the Chen's equality becomes δ(_M) = 2. and if M is assumed to be of constant scalar curvature, we classify M by two explicitely described immersions of S(^3) in S(^6). [C.D.V.Vl]. By assuming that the complementary distribution of a certain distribution of M is integrable. M is characterized in terms of a warped product of a minimal, totally real, non-totally geodesic surface immersion in S(^6). which lies linearly full in a totally geodesic S(^5) [C.D.V.V2]. Furthermore, with respect to totally real 3- dimensional submanifolds satisfying Chen's equality ([D.V]): if M is contained in a totally geodesic S(^5). then M can be classified in terms of complex curves in CP(^2)(4) lifted via the Hopf fibration. These submanifolds satisfy always Chen's equality. In case M lies linearly full in S(^6) and satisfies Chen's equality classification has been in terms of tubes of radius (^π)(_2) in the direction of the second normal space, over an almost complex curve. Finally, local converses of the last two results are proved.