We analyze the Dirac Laplacian of a one-parameter family of Dirac operators on a compact Lie group, which includes the Levi-Civita, cubic, and trivial Dirac operators. More specifically, we describe the Dirac Laplacian action on any Clifford module in terms of the action of the Lie algebra’s Casimir element on finitedimensional irreducible representations of the Lie group. Using this descriptio...

This paper addresses the general problem of generating smooth trajectories between an initial and a nal position and orientation. A functional depending on velocity and its higher derivatives involving a left invariant Riemannian metric on SE(3) is used to measure the smoothness of a trajectory. The problem of determining a smooth trajectory between two points is formulated as a variational pro...

We show that every Lie algebra or superLie algebra has a canonical braiding on it, and that in terms of this its enveloping algebra appears as a flat space with braided-commuting coordinate functions. This also gives a new point of view about q-Minkowski space which arises in a similar way as the enveloping algebra of the braided Lie algebra gl2,q. Our point of view fixes the signature of the m...

In this thesis we investigate the existence of complex-valued harmonic morphisms on Lie groups and homogeneous Hadamard manifolds. The Lie groups that we are interested in have a particular decomposition of their Lie algebra. This decomposition allows us to define harmonic morphisms to R, n ≥ 2. Any homogeneous Hadamard manifold is isometric to a solvable Lie group S with a left-invariant metri...

In this paper we construct negatively curved Einstein spaces describing gravitational waves having a solvegeometry wave-front (i.e., the wave-fronts are solvable Lie groups equipped with a left-invariant metric). Using the Einstein solvmanifolds (i.e., solvable Lie groups considered as manifolds) constructed in a previous paper as a starting point, we show that there also exist solvegeometry gr...

We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of R. In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S is preserved by the antipodal map. The stud...

Here, the left-hand side is the L-cohomology of M with respect to a (locally) invariant metric. Though it would be more natural to allow p = ∞ in Theorem 1, this is not generally possible (see (3.2.2)). On the other hand, there is a natural mapping H (∞)(M) → H • (p)(M) when p < ∞, because M has finite volume. The definition of M is recalled in (1.9). Theorem 1 can be viewed as an analogue of t...