Seminar Geometrija, vizuelizacija i obrazovanje sa primenama, 9. septembar 2010.


U četvrtak, 9. septembra 2010. u 17:15 časova u sali 301f  Matematičkog Instituta SANU, Knez Mihajlova 35, održaće se vanredni sastanak  Seminara.

Predavanje pod nazivom "Connections and Curvature in Differential Geometry" održaće Graham Hall, University of Aberdeen.


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.

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