Date & Time: Tuesday, September 23, 2008, 15:00-16:00.
Venue: Room 114
Title: Invariant Structures on Spaces of Geometric Objects
Speaker: Vikram Aithal, IIT Bombay
Abstract: The spaces of lines, circles, or triangles in the Euclidean plane, Sphere, or Hyperbolic plane, the non-singular conics in the projective plane, ... are some examples of "Spaces of Geometric Objects". Many times, these spaces are homogeneous spaces of Lie groups, or there are only finitely many orbits under the action of a Lie group. We consider the problem of existence of an invariant geometric structure, such as a (pseudo-)Riemannian metric, or a measure, under the action of a Lie group. In particular, we shall give a criterion for a G-invariant measure, when G is a Lie group. This issue is discussed in the well-known texts of Helgason and Santaló. But the criterion we obtain is neat and appears to be new.