An introduction to kinematic formulas (and why they can be studied on manifolds)

The kinematic formulas express the expected measure of the intersection of two convex bodies under random rigid motions as a sum of products of their intrinsic volumes. In the Euclidean setting, these formulas follow directly from Hadwiger’s characterization theorem and can be localized via Federer’s curvature measures. Between 2005 and 2007, Alesker laid the foundations of valuation theory on manifolds to extend integral geometry to the non-flat setting. After a brief introduction to the fundamental elements of classical integral geometry, we use tools coming from Alesker’s theory to study local kinematic formulas on an isotropic manifold (M, G), developing an algebraic operation that encodes this formulas in full generality. This work is part of my PhD thesis, supervised by Andreas Bernig at Goethe Universitat Frankfurt. The talk is part of the cycle of seminars ASK