Sala Seminari (Dip. Matematica)
Given an n-dimensional hyperbolic manifold M, it is reasonable to ask whether or not it can be realized as a totally geodesic, embedded submanifold of an (n+1)-dimensional hyperbolic manifold. If this is true, we say that M geodesically embeds. Determining whether or not a hyperbolic manifolds geodesically embeds is, in general, quite difficult.
However, if we restrict our attention to the case of arithmetic manifolds of simplest type, (a class of manifolds constructed using tools from number theory), we can show that, in many cases, they do indeed embed geodesically.
Let N/K be a normal tame extension of number fields and let G := Gal(N/K)
be its Galois group. It is common knowledge that, under these conditions,
[O_N] lies in the locally free class group Cl(O_K[G]). What can be said about
the existence of a normal integral basis for the extension N/K, by knowing
the structure of the group Cl(O_K[G])? In some cases this problem is easily
settled via cancellation law.
There is also a rank notion for locally free modules over orders. Which
In this talk I’ll describe the problem of filling submanifolds with topological or holomorphic disks. The case of geodesics on compact Riemannian surfaces with nonpositive scalar curvature will be treated. I will prove non existence of such disk filling, using several different tecniques. Two possible generalizations in higher dimension will be shown:
– the product of geodesics on the product of compact Riemannian surfaces with nonpositive scalar curvature does not admit a holomorphic disk filling;