Let G be a Lie group and K a compact subgroup of G. Then the homogeneous space G/K has an invariant Riemannian metric and an invariant volume form ΩG. Let M and N be compact submanifolds of G/K, and I(M ∩ gN) an “integral invariant” of the intersection M ∩ gN . Then the integral

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.

Lie groups appear in many fields fromMedical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ’s shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform sta...

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.

The real Jacobi group $G^J_n(\mathbb{R})$, defined as the semidirect product of Heisenberg ${\rm H}_n(\R)$ with symplectic ${\mr {Sp}}(n,\mathbb{R})$, admits a matrix embedding in $\text{Sp}(n+1,\mathbb{R})$. modified pre-Iwasawa decomposition $\rm{Sp}(n,\mathbb{R})$ allows us to introduce convenient coordinatization $S_n$ which for $G^J_1(\mathbb{R})$ coincides $S$-coordinates. Invariant one-f...