Morse-Bott functions on orthogonal groups

Abstract

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups O(n). We describe the critical loci of the quadratic trace function Tr(AXBX T ) and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of A and B. We give a simplified treatment of T. Frankel's analysis of the linear trace function on SO(n), as well as a combinatorial explanation of the relationship between the mod 2 Betti numbers of SO(n) and those of the Grassmannians G(2k, n) obtained from this analysis. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod 2 Betti numbers of SO(n).

Anchoring Sagittal Plane Templates in a Spatial Quadruped

“…Consider the gradient ∇ R Φ with respect to ω ∈ R 3 . Apply equation (3.1) from [29], recalling that y T Ry = Tr(yy T R):…”

“…B. The Hessian of Φ Lemma 3.2 from [29] gives an expression for the Hessian of Φ at any critical point as a symmetric bilinear form on so(n). Applying this result to (4) yields…”