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Register a new userGeneric immersions and totally real embeddings
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We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex 4k-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.
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An immersion of a smooth $n$-dimensional manifold $M to mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion $M to mathbb{R}^{TN(M)}$. In analogy with the totally skew embeddings studied by Ghomi and Tabachnikov, we find that totally nonparallel immersions are related to the generalized vector field problem, the immersion problem for real projective spaces, and nonsingular symmetric bilinear maps. Our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space $F_k(M)$ of k-tuples of distinct points of $M$; for example, k-regular embeddings, k-skew embeddings, k-neighborly embeddings, etc. Typically, a map satisfying one of these configuration space conditions induces some $S_k$-equivariant map on the configuration space $F_k(M)$ (or on a bundle thereof) and obstructions can be computed using Stiefel-Whitney classes. However, the existence problem for such conditions is relatively unstudied. Our main result is a Whitney-type theorem: every $n$-manifold $M$ admits a totally nonparallel immersion into $mathbb{R}^{4n-1}$, one dimension less than given by genericity. We begin by studying the local problem, which requires a thorough understanding of the space of nonsingular symmetric bilinear maps, after which the main theorem is established using the removal-of-singularities h-principle technique due to Gromov and Eliashberg. When combined with a recent non-immersion theorem of Davis, we obtain the exact value $TN(mathbb{R}P^n) = 4n-1$ when $n$ is a power of 2. This is the first optimal-dimension result for any closed manifold $M eq S^1$, for any of the recently-studied configuration space conditions.
Inspired by an article of R. Bryant on holomorphic immersions of unit disks into Lorentzian CR manifolds, we discuss the application of Cartans method to the question of the existence of bi-disk $mathbb{D}^{2}$ in a smooth $9$-dimensional real analytic real hypersurface $M^{9}subsetmathbb{C}^{5}$ with Levi signature $(2,2)$ passing through a fixed point. The result is that the lift to $M^{9}times U(2)$ of the image of the bi-disk in $M^{9}$ must lie in the zero set of two complex-valued functions in $M^{9}times U(2)$. We then provide an example where one of the functions does not identically vanish, thus obstructing holomorphic immersions.
Using the theory of totally real number fields we construct a new class of compact complex non-K{a}hler manifolds in every even complex dimension and study their analytic and geometric properties.
We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surface. In the case that the orbifold is a disk, there are some conditions. Our results generalize work of Calegari-Louwsma and resolve a conjecture of Calegari.
A totally real theta characteristic of a real curve is a theta characteristic which is linearly equivalent to a sum of only real points. These are closely related to the facets of the convex hull of the canonical embedding of the curve.
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