**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.