Seminar on Geometry, visualisation and education with applications, September 9th, 2010.
- 31. August, 2010
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Prof Graham Hall, University of Aberdeen, will give a lecture on Thursday, September 9rd, 2010. The lecture will be held at Mathematical Institute, Knez Mihailova street, 35, room 301f at 17:15h.
Lecture's title: "Connections and Curvature in Differential Geometry" 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.