Non-vanishing complex vector fields and the Euler characteristic

The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified.Comment: 2 page

A Conjecture of Trautman

In 1998 the physicist Andre Trautman conjectured that a three-dimensional CR manifold is locally realizable if and only if its canonical bundle admits a closed nowhere zero section. We review the relevant definitions, give the physical context, and outline an earlier result which established a weak version of the Conjecture

Locally CR Spherical Three Manifolds

Every open and orientable three manifold has a CR structure which is locally equivalent to the standard CR structure on $S^3$

CR structures on open manifolds

We show that the vanishing of the higher dimensional homology groups of a manifold ensures that every almost CR structure of codimension $k$ may be homotoped to a CR structure. This result is proved by adapting a method due to Haefliger used to study foliations (and previously applied to study the relation between almost complex and complex structures on manifolds) to the case of (almost) CR structures on open manifolds

The ∂‾\overline\partial-cohomology groups, holomorphic Morse inequalities, and finite type conditions

We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if and only if for any positive constant $C$, the number of eigenvalues of the $\overline\partial$-Neumann Laplacian less than or equal to $Ck$ grows polynomially as $k$ tends to infinity.Comment: 27 page

Optimality for totally real immersions and independent mappings of manifolds into C^N

The optimal target dimensions are determined for totally real immersions and for independent mappings into complex affine spaces. Our arguments are similar to those given by Forster, but we use orientable manifolds as far as possible and so are able to obtain improved results for orientable manifolds of even dimension. This leads to new examples showing that the known immersion and submersion dimensions for holomorphic mappings from Stein manifolds to affine spaces are best possible.Comment: 12 page

Left-invariant CR structures on 3-dimensional Lie groups

The systematic study of CR manifolds originated in two pioneering 1932 papers of \'Elie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on $\mathrm{SU}_2 = S^3$, deforming the standard `spherical' structure. In this paper, mostly expository, we illustrate and clarify Cartan's results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that $\mathrm{SL}_2(\mathbb{R})$ admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of $\mathrm{SL}_2(\mathbb{R})$ provide CR embedding or immersions of these structures. The same methods apply to all other three-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for $\mathrm{SU}_2$, the Heisenberg group, and the Euclidean group.Comment: 23 pages, 1 figur

The cohomology of left-invariant CR structures of hypersurface type on compact Lie groups

Pittie [Pit88] proved that the Dolbeault cohomology of all left-invariant complex structures on compact Lie groups can be computed by looking at the Dolbeault cohomology induced on a conveniently chosen maximal torus. We use the algebraic classification of left-invariant CR structures of maximal rank on compact Lie groups [CK04] to generalize Pittie's result to left-invariant Levi-flat CR structures of maximal rank on compact Lie groups.Comment: 19 pages, 1 tabl

GENERIC SYSTEMS OF CO-RANK ONE VECTOR DISTRIBUTIONS

Abstract. 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 Poincaré dual of the characteristic set of the associated system of linear partial differential equations. Let M n be a compact manifold. We study generic sub-bundles of co-rank one. In defining a condition of genericity, we are guided by the needs of the theory of linear partial differential equations and by the theory of generic functions as initiated by Whitney. We use V to denote a sub-bundle of CTM of co-rank one. Such a sub-bundle defines a map ΦV: M↩ → PCT ∗ M, where the image of a point p is the line of one-forms which annihilate Vp. We are interested in points where this line is generated by a real form, so we set Σ(V)={p ∈ M:ΦV(p) ∈ PT ∗ M}. This is the characteristic set of the system of first order partial differential operators corresponding to any local basis for V. In particular, the system is elliptic outside of Σ. As we have just done, we freely write Σ for Σ(V). All manifolds, bundles, and maps are taken to be of class C ∞. Definition 1. V is an almost Whitney structure if (1) ΦV (M) intersects PT∗M transversely. It follows that Σ is a smooth curve with a finite number of components. (2) ω(τ) has only simple zeroes when τ is a nonzero section of T Σandωis a one form such that ΦV =[ω] onΣ. Remark 0.1. We follow the convention of introducing“almost ” so that we can drop it once we require formal integrability, i.e. involutivity. Thus V is a Whitney structure if in addition (3) [V,V] ⊂ V.(See§3.) Example. Let M = X × S1,whereXis a compact two-dimensional manifold (or, more generally, let M be a circle bundle over X with orientable fiber). Let ξ be a real nondegenerate vector field on X with {p ∈ X: ξ(p) =0} = {p1,...,pN}. That is, at each pj the linear approximation to ξ is given by a matrix whose determinant is Received by the editors September 8, 2004

NON-VANISHING COMPLEX VECTOR FIELDS AND THE EULER CHARACTERISTIC

Abstract. 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. One of the oldest and most basic results in global differential topology relates the topology of a manifold to the zeros of its vector fields. Let M be a compact and orientable manifold and let χ(M) denote its Euler characteristic. Here is the simplest statement of this relation. (1) If there is a global nowhere zero vector field on M then χ(M) = 0. This of course is for a real vector field (that is, for a section M → TM). On the other hand, it is easy to see that any manifold admits a nowhere zero complex vector field. (A complex vector field is a section M → C ⊗ TM). This can be seen most simply by observing that a generic perturbation of any section, even the zero section itself, must be everywhere different from zero. It is natural to seek a condition on a nowhere zero complex vector field which would again imply χ(M) = 0. Curiously, a trivial restatement of (1) leads to such a condition. Let g be any Riemannian metric on M. (2) Let v: M → TM be a global vector field on M. If the Riemannian metric g(v, v) is never zero, then χ(M) = 0. Here is the condition for complex vector fields. Theorem 1. Let v: M → C ⊗ TM be a global vector field on M. If the bilinear form g(v,v) is never zero, then χ(M) = 0. Here g is extended to complex vector fields by taking g(v, w) to be complex linear in each argument; for v = ξ + iη we have g(v, v) = g(ξ, ξ) − g(η, η) + 2ig(ξ, η). Proof. We show that if g(v, v) ̸ = 0 then v can be deformed to a nowhere zero real vector field. So the Euler characteristic would be zero, according to (1). We decompose M as M = A+ ∪ B ∪ A− where g(ξ, ξ)> g(η, η) on A+, the opposite inequality holds on A−, and equality holds on B. We assume for now that B is not empty. Note that ξ is nowhere zero in A+ and η is nowhere zero in A−. Further, since g(v, v) is never zero, we hav