Seminar for Geometry, education and visualization with applications
MATEMATIČKI INSTITUT SANU
Seminar geometriju, obrazovanje i vizualizaciju sa primenama
PLAN RADA ZA SEPTEMBAR 2010.
ČETVRTAK, 09.09.2010. u 17 sati, sala 301f, MI
Graham Hall, University of Aberdeen
Connections and Curvature in Differential Geometry
Apstrakt: In this talk I will try to indicate precisely the relationships between the various ways that the curvature of a manifold can manifest itself. Let M be a manifold of dimension n admitting a connection D which is the Levi-Civita connection of a metric g on M of arbitrary signature. The curvature, fixed by g, shows itself in many ways; by the curvature tensor Riem derived from D, by the holonomy group of D, by the unparametrised geodesics arising from D (the so-called, projective structure), by the Weyl conformal tensor, C, by the Weyl projective tensor, W, by the sectional curvature of Riem and g and, doubtless, many others. Clearly, g determines D, Riem, C, W and the sectional curvature. It is then interesting to ask how many other relationships there are between these structures and, in particular, to what extent the original metric g can be recovered from each of them. I will show that there are many interesting links between them and, in the situation when the dimension of M is small, they can be quite precisely stated. On the other hand, not all such relations between them are as convenient as one might like and I will indicate this by means of examples. Such problems are, of course, interesting for differential geometers. In addition, the situation when M has dimension 4 and g is of Lorentz signature, is interesting for general relativity theory, especially the connection between the projective structure (more precisely, the unparametrised, timelike geodesics on M) and the Newton-Einstein principle of equivalence.
ČETVRTAK, 30.09.2010. u 17 sati, sala 301f, MI SANU
REAL SUBMANIFOLDS OF CODIMENSION 2 OF A COMPLEX SPACE FORM
Apstrakt: Let M be a real submanifold of a complex manifold M and J be the natural almost complex structure of M. For x 2 M, we call the subspace Hx(M) = JTx(M) \ Tx(M) of the tangent space Tx(M) the holomorphic tangent space of M. If the holomorphic tangent space has con- stant dimension with respect to x 2 M, the submanifold is called a CR submanifold and the constant complex dimension is called the CR dimension of M. It is well known that an n-dimensional real hypersurface of a complex manifold is a CR sub- manifold of CR dimension n?1 2 . We consider now real submanifolds of codimension 2 of a complex manifold. Then contrary to real hypersurfaces, the submanifolds are something complicated. They are not only CR submanifolds of CR dimension n?2 2 , but also some other cases. For example a complex hyper- surface is a real submanifold of codimension 2. Moreover there exists a submanifold which is not CR submanifold. However, to investigate even dimensional real submanifolds of complex manifold, codimension 2 case is fundamental. We investigate real submanifolds of codimension 2 of a complex manifold under the condition that h(FX; Y ) + h(X; FY ) = 0 (¤) and obtained the following results. Theorem 1. If a complex hypersurface M of a KAahler maniold M satis?es the condition (*), M is a totally geodesic submanifold. Theorem 2. Let M be a non Euclidean complex space form. If a real submanifold M of codimension 2 satis?es the condition (*), then one of the following holds. (1) M is a totally geodesic complex hypersurface. (2) M is a CR submanifold of CR dimension n?2 2 with ? = 0. Theorem 3. Let M be a real submanifold of codimension 2 of coplex Euclidean space Cn+2 2 which satis?es the condition (*). Then M is one of the following: (1) M is an n-dimensional Euclidean space En, (2) M is an n-dimensional sphere Sn, (3) M is a product of an even dimensinal sphere with Euclidean sspace Sr ? En?r (4) M is a CR submanifold of CR dimension n?2 2 with ? = 0. Where ? is a function de?ned on the submanifold M.
Sednice seminara odrzavaju se u zgradi Matematickog instituta SANU, Knez-Mihailova 36, na trecem spratu u sali 301f.
Rukovodilac Seminara dr Miroslava Antic