Any smooth connected non-orientable manifold is equipped with a real line bundle of order two. Various structures which are defined only on oriented manifolds extend to non-orientable manifolds once they are twisted by this line bundle of order two. Our aim is to develop this theme....
The class of complex Hantzsche-Wendt manifolds was defined in [2] as a complex analogue to the class of Hantzsche-Wendt manifolds. The latter is a class of flat orientablen-dimensional manifolds with holonomy group, a generalisation of the original Hantzsche-Wendt manifold which is the unique...
In addition, the following result turned out to be useful for our purposes Theorem 2.1 [12, Theorem 2.1] Suppose that M is a compact orientable hypersurface embedded in a compact Riemannian manifold N . If the Ricci curvature of N is bounded below by a positive constant k, then 2μ1 >...
Let M be a manifold of dimension n which is immersible in Rn+1 . Then T c M is trivial if M is orientable, or M is non-orientable with H 2 (M , Z) = 0. In fact, Gromov proved a stronger result that M admits an exact Lagrangian immersion in Euclidean space when M is ...
In particular, the complex span of an almost complex manifold M2n is defined to be the complex span of the complex n-dimensional vector bundle whose underlying real bundle is (isomorphic to) the tangent bundle T(M). Of course, then the real span of M is at least twice the complex span...
boundary [W1]. If the boundary is S 3 then of course the Artin presentation presents the trivial group. Even in this case the Artin presentation already en- codes all of the smooth structure of the 4-manifold. Thus, it makes sense to ask ...
Suppose N is a compressible boundary component of a compact irreducible orientable 3-manifold M, and (Q,?Q) ? (M,?M) is an orientable properly embedded essential surface in M, some essential component of which is incident to N and no component is a disk. Let V and Q denote ...
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....
A Morse function f on a manifold with corners M allows the characterizationof the Morse data for a critical point by the Morse index. In fact, a modifiedgradient flow allows a proof of the Morse theorems in a manner similar to thatof classical Morse theory. It follows that M is homotopy...
As an application of this complex it is shown that for a closed orientable 3-manifold, and any of its Heegaard splittings, one can give an algorithm to decide whether the manifold contains a 2-sided, non-separating, closed incompressible surface (Theorem 1.1).Ningthoujam Jiban Singh...