Real Projective Space Research Articles

Half-space results for generalized linear Weingarten hypersurfaces immersed in the real projective space

Half-space results for generalized linear Weingarten hypersurfaces immersed in the real projective space

Regularity of limit sets of Anosov representations

AbstractIn this paper, we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a ‐submanifold of the real projective space for some . We also calculate the optimal value of in terms of the eigenvalue data of the Anosov representation.

On the Algebraic Geometry of Multiview

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras.

Hofer–Zehnder capacity of disc tangent bundles of projective spaces

AbstractWe compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.

On multiplicative spectral sequences for nerves and the free loop spaces

We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a category internal to the category of topological spaces. By applying the machinery to a Borel construction, we explicitly determine the mod p cohomology algebra of the free loop space of the real projective space for each odd prime p. This example is emphasized as an important computational case. Moreover, we represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold M with differential forms on the universal cover of M via Chen's iterated integral map.

Average Degree of the Essential Variety

The essential variety is an algebraic subvariety of dimension 5 in real projective space RP8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R\\mathrm P^{8}$$\\end{document} which encodes the relative pose of two calibrated pinhole cameras. The 5-point algorithm in computer vision computes the real points in the intersection of the essential variety with a linear space of codimension 5. The degree of the essential variety is 10, so this intersection consists of 10 complex points in general. We compute the expected number of real intersection points when the linear space is random. We focus on two probability distributions for linear spaces. The first distribution is invariant under the action of the orthogonal group O(9)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{O}(9)$$\\end{document} acting on linear spaces in RP8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R\\mathrm P^{8}$$\\end{document}. In this case, the expected number of real intersection points is equal to 4. The second distribution is motivated from computer vision and is defined by choosing 5 point correspondences in the image planes RP2×RP2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R\\mathrm P^2\ imes \\mathbb R\\mathrm P^2$$\\end{document} uniformly at random. A Monte Carlo computation suggests that with high probability the expected value lies in the interval (3.95-0.05,3.95+0.05)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3.95 - 0.05,\\ 3.95 + 0.05)$$\\end{document}.

On the intertwining differential operators from a line bundle to a vector bundle over the real projective space

We classify and construct SL(n,R)-intertwining differential operators D from a line bundle to a vector bundle over the real projective space RPn−1 by the F-method. This generalizes a classical result of Bol for SL(2,R). Further, we classify the K-type formulas for the kernel Ker(D) and image Im(D) of D. The standardness of the homomorphisms φ corresponding to the differential operators D between generalized Verma modules is also discussed.

Convex co‐compact representations of 3‐manifold groups

AbstractA representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples.

Roudneff’s Conjecture in Dimension 4

J.-P. Roudneff conjectured in 1991 that every arrangement of n ≥ 2 d + 1 ≥ 5 pseudohyperplanes in the real projective space P d has at most ∑ i = 0 d − 2 ( n − 1 i ) complete cells (i.e., cells bounded by each hyperplane). The conjecture is true for d = 2, 3 and for arrangements arising from Lawrence oriented matroids. Our main contribution is to show the validity of Roudneff’s conjecture for d = 4. Moreover, based on computational data we show that for d ≤ 4 and n ≤ 2 d + 1 the maximum number of complete cells is only obtained (up to isomorphism) by cyclic arrangements.

Index of Minimal Hypersurfaces in Real Projective Spaces

Abstract We prove that for an embedded unstable one-sided minimal hypersurface of the $(n+1)$-dimensional real projective space, the Morse index is at least $n+2$, and this bound is attained by the cubic isoparametric minimal hypersurfaces. We also show that there exist closed embedded two-sided minimal surfaces in the 3-dimensional real projective space of each odd index by computing the index of the Lawson surfaces.

Topological Complexity and LS-Category of Certain Manifolds

The Lusternik–Schnirelmann category and topological complexity are important invariants of topological spaces. In this paper, we calculate the Lusternik–Schnirelmann category and topological complexity of products of real projective spaces and their wedge products by using cup and zero-cup length. Also, we will find the topological complexity of R P 2 k + 1 by using the immersion dimension of R P 2 k + 1 .

Real hypersurfaces with Ricci–Bourguignon soliton in the complex two-plane Grassmannians

The study of Ricci-Bourguignon soliton on real hypersurfaces in the complex two-plane Grassmannian [Formula: see text] is first investigated. It is proved that there exists a shrinking Ricci-Bourguignon soliton on a Hopf real hypersurface [Formula: see text] in [Formula: see text] by using pseudo-anticommuting Ricci tensor. Moreover, we have proved that there does not exist a nontrivial gradient Ricci-Bourguignon soliton ([Formula: see text]) on real hypersurfaces with isometric Reeb flow in the complex two-plane Grassmannian [Formula: see text]. Among the class of contact hypersurface in [Formula: see text], we also prove that there does not exist a nontrivial gradient Ricci-Bourguignon in [Formula: see text] over the totally geodesic and totally real quaternionic projective space [Formula: see text] in [Formula: see text], [Formula: see text].

Geometry of knots in real projective 3-space

This paper discusses some geometric ideas associated with knots in real projective 3-space [Formula: see text]. These ideas are borrowed from classical knot theory. Since knots in [Formula: see text] are classified into three disjoint classes: affine, class-[Formula: see text] non-affine and class-[Formula: see text] knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in [Formula: see text]. We then study the notion of companionship of knots in [Formula: see text] and using it we provide geometric criteria for a knot to be affine. We also define a notion of “genus” for knots in [Formula: see text] and study some of its properties. We prove that this genus detects knottedness in [Formula: see text] and gives some criteria for a knot to be affine and of class-[Formula: see text]. We also prove a “non-cancellation” theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class-[Formula: see text] non-affine knots with genus [Formula: see text]. Thus we highlight that, [Formula: see text] admits a knot theory with a truly different flavor than that of [Formula: see text] or [Formula: see text].

Isoperimetry and volume preserving stability in real projective spaces

Isoperimetry and volume preserving stability in real projective spaces

Some half-space theorems in the real projective space

Some half-space theorems in the real projective space

Simultaneous linearization of diffeomorphisms of isotropic manifolds

Suppose that M is a closed isotropic Riemannian manifold and that R_1,\dots ,R_m generate the isometry group of M . Let f_1,\dots ,f_m be smooth perturbations of these isometries. We show that the f_i are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475–505 (2007)] from S^n to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.

The first and second widths of the real projective space

In this paper, we deal with the first and second widths of the real projective space R P n \mathbb {RP}^{n} , for n n ranging from 4 4 to 7 7 , and for this we used some tools from the Almgren-Pitts min-max theory. In a recent paper, Ramirez-Luna computed the first width of the real projective spaces, and, at the same time, we obtained optimal sweepouts realizing the first and second widths of those spaces.

Holography for mathcal{N} = 4 on mathbbm{RP} 4

We propose a holographic description of mathcal{N} = 4 super Yang-Mills on the four-dimensional real projective space mathbbm{RP} 4. We first construct the dual background in the framework of five-dimensional mathcal{N} = 8 gauged supergravity, and then uplift it to a new one-half BPS solution of type IIB supergravity. A salient feature of our solution is the presence of a bulk naked singularity whose local behavior resembles that of an O1− plane in flat space.

Free rank of symmetry of products of Dold manifolds

AbstractDold manifolds $P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and $ \mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to ${\prod_{i=1}^{k} P(2m_i,n_i)}$, then $G\cong (\mathbb{Z}_2)^l$ for some $l\leq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i, ni is even and mi is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions.

Boundary measurement and sign variation in real projective space

We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space.We find the spaces to be PL manifolds with boundary which are homotopy equivalent to another real projective space of smaller dimension. One generalization makes use of sign variation while the other uses boundary measurement. Spaces arising from boundary measurement are shown to admit Cohen–Macaulay triangulations.