In this paper we give a method for constructing complex valued harmonic morphisms in some pseudo-Riemannian manifolds using a parametrization of isotropic subbundles of the complexified tangent bundle. As a result we construct the first known examples of complex valued harmonic morphisms in real hyperbolic spaces of even dimension not equal to 4 which do not have totally geodesic fibres.

We study a natural Dirac operator on a Lagrangian submanifold of a Kähler manifold. We first show that its square coincides with the Hodge de Rham Laplacian provided the complex structure identifies the Spin structures of the tangent and normal bundles of the submanifold. We then give extrinsic estimates for the eigenvalues of that operator and discuss some examples. Mathematics Subject Classif...

We establish that, for every hyperbolic orbifold of type (2, q,∞) and for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent bundle is left-handed. This implies that the link formed by every collection of periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii) is a fibered link. We also prove similar results for the torus with any flat metric. Besid...

A diffeomorphism f : M → M on a compact manifold M is partially hyperbolic if there exists a continuous, nontrivial Df -invariant splitting TxM = E s x ⊕ E c x ⊕ E u x , x ∈ M of the tangent bundle such that the derivative is a contraction along E and an expansion along E, with uniform rates, and the behavior of Df along the center bundle E is in between its behaviors along E and E, again by a ...

We study dynamics in a neighborhood of a nonhyperbolic fixed point or an irreducible homoclinic tangent point. General type conditions for the existence of infinite sets of periodic points are obtained. A new method, based on the study of the dynamics of center disks, is introduced. Some results on shadowing near a non-hyperbolic fixed point of a homeomorphism are obtained.

This course focuses on so-called totally characteristic, or b-(pseudo)differential operators. These are operators on manifolds M with boundaries or corners, though they can also arise on complete manifolds without boundary after one appropriately compactifies or bordifies them, as we discuss below. We study both elliptic and hyperbolic operators, with the Laplacian or the d’Alembertian of a b-m...

We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When B is non-trapping, the magneti...

In this talk we describe some recent developments in microlocal analysis that have led to advances in understanding problems such as wave propagation, the Laplacian on asymptotically hyperbolic spaces and the meromorphic continuation of the dynamical zeta function for Anosov flows.

In this paper, hyperbolic spinor representations of space curves are studied according to the q-frame in E_1^3. The formulations calculated for spacelike and timelike tangent vector cases Moreover, relationships equations between Frenet frame Lorentz expressed. results supported with some theorems.