Submanifolds associated with graphs
by
Alfonso Carriazo
University of Sevilla, Spain
Coauthors: Luis M. Fernández (University of Sevilla)

Poster

An important research subject in the theory of submanifolds of an almost Hermitian manifold is the study of a submanifold from the point of view of its behaviour with respect to the almost complex structure. Many submanifolds are characterized by such an homogeneous behaviour: holomorphic submanifolds, totally real submanifolds, CR-submanifolds, and, more recently, slant, semi-slant or bi-slant submanifolds. In all these cases, we can find a graph closely related to the algebraic structure on the submanifold.

In this contribution, we generalize the above situation by defining submanifolds associated to graphs. Basically, we first construct a graph related to the tangent space at an arbitrary point of a submanifold, and then, we say that this submanifold is associated to the graph if it can be differentably extended to every point of the submanifold, in a certain way.

We show some results about the possibility of a graph to be associated to a submanifold and we use them to characterize CR-submanifolds by means of trees (connected graphs without cycles). Finally, we study and characterize submanifolds associated to graphs in a four-dimensional almost Hermitian manifold. We also offer some open problems on this subject.

We think that the appropriate analysis of the association between these two different objects (the submanifolds from the Complex Geometry and the graphs from the Discrete Mathematics) will provide us with a new tool which can be useful to study the general problem of the classification of submanifolds. Afterwards, it establishes a new link between two traditionally remote research areas.

Date received: November 25, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-03.