“Submanifolds of Nearly Kaehler spaces”
The proposed project is about the study of differential geometry. The objects that we study are called manifolds and are more complicated than the points, lines and curves from basic geometry. They could be seen as surfaces of higher dimensions in different ambient spaces. For example, one could think of the blackboard in the classroom as a flat surface (the blackboard) in the 3-dimensional Euclidean space (the classroom), which is the ambient space. If one slightly bent the blackboard without breaking it, one could obtain a more general surface. In geometry, the surface doesn't always have 2 dimensions, nor the ambient space 3, but higher arbitrary dimensions. Endowed with particular mathematical properties, these manifolds may correspond to objects of high interest in classical mechanics or advanced physics, for example. Mathematics offers a whole set of tools in order to be able to study these objects. By means of analysis, algebra, topology, etc., we can do computations on the manifolds, in order to understand their properties and describe them by formulas. We are also interested about relations between such objects. Manifolds that stay inside other manifolds are called submanifolds (one could think about the trace of a river through a field as a curve (a submanifold) in a plane (a manifold)). A particular class of submanifolds (called 'Lagrangian') that live inside other ambient manifolds called 'nearly Kaehler manifolds' are of central interest in the project.