Descripción
**Hora de inicio:** 11.00.
Se impartirá como parte del curso LAAGRANGIAN SUBMANIFOLDS del Programa de Doctorado Matemáticas.
Abstract:
A tensor T is called isotropic if T(v,...,v) is independent of the choice of unit length vector v. The first result about Lagrangian submanifolds admitting an isotropic tensor was due to Naitoh who classified isotropic parallel Lagrangian submanifolds of complex space forms. Later results are due to Montiel, Urbano and Ejiri. In both cases the tensor T(X,Y,Z,W)=<h(X,Y),h(Z,W)>, where h is the second fundamental form of the Lagrangian immersion. The classification result in this case either gives the parallel hypersurfaces of Naitoh or a special class of H umbilical Lagrangian submanifolds.
Of course the same question can also be asked for other geometric tensors like T(X,Y,Z,W)=<∇h(X,Y,Z),JW> or T(X,Y,Z,W,U,V)= <∇h(X,Y,Z),∇h (W,U,V)>.
The first condition actually can be used to characterize the Whitney spheres, together with the parallel Lagrangian submanifolds. Whereas the second one is more difficult to treat and so far a classification of it is only known in dimension 3.
Of course another possibility to generalize the previous results is to look at Lagrangian submanifolds of inefinite complex space forms. As a basic ingredient in the positive definite case which is the choice of a canonical frame based on the choice of e_1 as a vector on which a certain function on a compact set attains an absolute maximum breaks down in the indefinite case; new methods need to be developed.
Moreover, the above developped techniques can also be used to study some submanifolds in affine differential geometry.
The results in these lectures are based on work in progress with F. Dillen, H.Li and X. Wang for Lagrangian submanifolds and with O.Birembaux and M. Djoric for affine differential geometry.