Howard Jacobowitz

Global Mizohata structures

The Mizohata partial differential operator is generalized to global structures on compact two-dimensional manifolds. A generalization of the Hopf Theorem on vector fields is used to show that a first integral can exist if and only if the genus is even. The Mizohata structures on the sphere are classified by the diffeomorphism group of the circle modulo the Moebius subgroup and a necessary and sufficient condition, expressed in terms of the associated diffeomorphism, is given for the existence of a first integral.

Curvature operators on the exterior algebra1

Properties of the exterior algebra of a vector space are used to investigate the curvature operator of a Riemannian manifold. Induced inner products and linear maps are used to establish results about the Euler characteristic of a compact manifold. An open problem about the decomposition of operators on A 2 V is discussed. This problem arises in the study of the codimension needed for isometric embeddings. A new algebraic consequence of the first Bianchi identities is established.

Sub-bundles of the complexified tangent bundle

We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.

Geometric Sensitivity of a Pinhole Collimator

Geometric sensitivity for single photon emission computerized tomography (SPECT) is given by a double integral over the detection plane. It would be useful to be able to explicitly evaluate this quantity. This paper shows that the inner integral can be evaluated in the situation where there is no gamma ray penetration of the material surrounding the pinhole aperature. This is done by converting the integral to an integral in the complex plane and using Cauchys theorem to replace it by one which can be evaluated in terms of elliptic functions.

Non-vanishing complex vector fields and the Euler characteristic

Every manifold admits a nowhere vanishing complex vector field. If, however, the manifold is compact and orientable and the complex bilinear form associated to a Riemannian metric is never zero when evaluated on the vector field, then the manifold must have zero Euler characteristic.

b-Completion of pseudo-Hermitian manifolds

We study the interrelation among pseudo-Hermitian and Lorentzian geometry as prompted by the existence of the Fefferman metric. Specifically for any nondegenerate Cauchy–Riemann manifold M we build its b-boundary ˙ M .T his arises as a S 1 quotient of the b-boundary of the (total space of the canonical circle bundle over M endowed with the) Fefferman metric. Points of ˙ M are shown to be endpoints of b-incomplete curves. A class of inextensible integral curves of the Reeb vector on a pseudo-Einstein manifold is shown to have an endpoint on the b-boundary provided that the horizontal gradient of the pseudoHermitian scalar curvature satisfies an appropriate boundedness condition. Dedicated to the memory of Stere Ianus ¸ 4

Maps into complex space

If the dimension of M is denoted by 2k - 1 or 2k, then a generic map F: M → C k satisfies dF 1 A... A dF k ≠0, while in certain cases there is no map F: M → C k+1 that satisfies dF 1 A... A dF k+1 ≠ 0.

Generic systems of co-rank one vector distributions

This paper studies a generic class of sub-bundles of the complexified tangent bundle. Involutive, generic structures always exist and have Levi forms with only simple zeroes. For a compact, orientable three-manifold the Chern class of the sub-bundle is mod 2 equivalent to the Poincare dual of the characteristic set of the associated system of linear partial differential equations.