Real Projective Space Research Articles

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.

Stolarsky’s invariance principle for projective spaces

We show that Stolarsky’s invariance principle, known for point distributions on the Euclidean spheres, can be extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane.

Cheeger constants, structural balance, and spectral clustering analysis for signed graphs

We introduce a family of multi-way Cheeger-type constants {hkσ,k=1,2,…,n} on a signed graph Γ=(G,σ) such that hkσ=0 if and only if Γ has k balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, those measures of bipartiteness introduced by Desai-Rao, Trevisan, Bauer–Jost, respectively, on unsigned graphs, and the frustration index (originally called the line index of balance by Harary) on signed graphs. We further unify the (higher-order or improved) Cheeger and dual Cheeger inequalities for unsigned graphs as well as the underlying algorithmic proof techniques by establishing their corresponding versions on signed graphs. In particular, we develop a spectral clustering method for finding k almost-balanced subgraphs, each defining a sparse cut. The proper metric for such a clustering is the metric on a real projective space. We also prove estimates of the extremal eigenvalues of signed Laplace matrix in terms of number of signed triangles (3-cycles).

Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous spaces. It is also shown that, for a symmetric and continuous matrix function with compact support, if it makes an isotropic covariance matrix function in the Euclidean space, then it makes an isotropic covariance matrix function on the sphere or the real projective space.

Topology of real Schläfli six-line configurations on cubic surfaces and in ℝℙ³

A famous configuration of 27 lines on a non-singular cubic surface in P 3 \mathbb {P}^3 contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case of real cubic surfaces from a topological viewpoint, as configurations of six disjoint lines in the real projective 3-space, and show that the condition that they lie on a cubic surface implies a very special property of homogeneity. This property distinguishes them in the list of 11 deformation types of configurations formed by six disjoint lines in R P 3 \mathbb {RP}^3 .

Lower and upper bounds for the waists of different spaces

We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space. We establish waist-type results in terms of the Hausdorff measure.

On the Number of Flats Tangent to Convex Hypersurfaces in Random Position

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287–300, 2006; Bürgisser and Lerario, in J Reine Angew Math, https://doi.org/10.1515/crelle-2018-0009, 2018; Megyesi and Sottile, in Discrete Comput Geom 33(4):617–644, 2005; Megyesi et al., in Discrete Comput Geom 30(4):543–571, 2003; Sottile and Theobald, in Trans Am Math Soc 354(12):4815–4829, 2002; Proc Am Math Soc 133(10):2835–2844, 2005; in: Goodman et al., in Surveys on discrete and computational geometry. AMS, Providence, 2008) we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view (here by “convex hypersurfaces” we mean that these hypersurfaces are boundaries of convex sets). More precisely, we say that smooth convex hypersurfaces $$X_1, \ldots , X_{d_{k,n}}\subset {\mathbb {R}}\text {P}^n$$, where $$d_{k,n}=(k+1)(n-k)$$, are in random position if each one of them is randomly translated by elements $$g_1, \ldots , g_{{d_{k,n}}}$$ sampled independently from the orthogonal group with the uniform distribution. Denoting by $$\tau _k(X_1, \ldots , X_{d_{k,n}})$$ the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that $$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})={\delta }_{k,n} \cdot \prod _{i=1}^{d_{k,n}}\frac{|\Omega _k(X_i)|}{|\text {Sch}(k,n)|}, \end{aligned}$$where $${\delta }_{k,n}$$ is the expected degree from [6] (the average number of k-flats incident to $$d_{k,n}$$ many random $$(n-k-1)$$-flats), $$|\text {Sch}(k,n)|$$ is the volume of the Special Schubert variety of k-flats meeting a fixed $$(n-k-1)$$-flat (computed in [6]) and $$|\Omega _k(X)|$$ is the volume of the manifold $$\Omega _k(X)\subset \mathbb {G}(k,n)$$ of all k-flats tangent to X. We give a formula for the evaluation of $$|\Omega _k(X)|$$ in terms of some curvature integral of the embedding $$X\hookrightarrow {\mathbb {R}}\text {P}^n$$ and we relate it with the classical notion of intrinsic volumes of a convex set: $$\begin{aligned} \frac{|\Omega _{k}(\partial C)|}{|\text {Sch}(k, n)|}=4V_{n-k-1}(C),\quad k=0, \ldots , n-1. \end{aligned}$$As a consequence we prove the universal upper bound: $$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})\le {\delta }_{k, n}\cdot 4^{d_{k,n}}. \end{aligned}$$Since the right hand side of this upper bound does not depend on the specific choice of the convex hypersurfaces, this is especially interesting because already in the case $$k=1, n=3$$ for every $$m>0$$ we can provide examples of smooth convex hypersurfaces $$X_1, \ldots , X_4$$ such that the intersection $$\Omega _1(X_1)\cap \cdots \cap \Omega _1(X_4)\subset \mathbb {G}(1,3)$$ is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions.

Rigidity of convex divisible domains in flag manifolds

In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of $p$-planes in $\mathbb{R}^{2p}$ when $p > 1$. Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group SO$(p, p)$.

Zeta determinant of the Laplacian on the real projective spaces

We give a closed form evaluation of the zeta determinant of the Laplace operator on spheres and projective spaces that clearly describes the arithmetic structure of this number. All the factors in the final formula for the determinant are easily computable.

Indices of a Finitistic Space with Mod 2 Cohomology ℝPn × $$\mathbb{S}^2$$ S 2

Let G = ℤ2 act freely on a finitistic space X with mod 2 cohomology ring isomorphic to the product of a real projective space and 2-sphere $$\mathbb{S}^2$$ . In this paper, we determine the Conner and Floyd’s mod 2 cohomology index and the Volovikov’s numerical index of X. Using these indices, we discuss the nonexistence of equivariant maps $$X\rightarrow\mathbb{S}^n$$ and $$\mathbb{S}^n\rightarrow{X}$$ . The covering dimensions of the coincidence sets of continuous maps X → ℝk are also determined.

The algebraic transfer for the real projective space

A chain-level representation of the Singer transfer for any left A-module is constructed. We prove that the image of the Singer transfer Tr⁎RP∞ for the infinite real projective space is a module over the image of the transfer Tr⁎ for the sphere. Further, the algebraic Kahn–Priddy homomorphism is an epimorphism from ImTr⁎RP∞ onto ImTr⁎ in positive stems. The indecomposable elements hˆi for i≥1 and cˆi, dˆi, eˆi, fˆi, pˆi for i≥0 are detected, whereas the ones gˆi for i≥1 and Dˆ3(i), pˆi′ for i≥0 are not detected by the Singer transfer Tr⁎RP∞. This transfer is shown to be not monomorphic in every positive homological degree. The transfer behavior is also investigated near “critical elements”. We prove that Kameko's squaring operation on the domain of Tr⁎RP∞ is eventually isomorphic. This phenomenon leads to the so-called “stability” of the Singer transfer for the infinite real projective space under the iterated squaring operation.

A characterization of Clifford parallelism by automorphisms

Betten and Riesinger have shown that Clifford parallelism on real projective space is the only topological parallelism that is left invariant by a group of dimension at least 5. We improve the bound to 4. Examples of different parallelisms admitting a group of dimension 3 are known, so 3 is the "critical dimension".

POINT DISTRIBUTIONS IN TWO‐POINT HOMOGENEOUS SPACES

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of -designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

Invariants of the $\mathbb{Z}_2$ orbifolds of the Podleś two spheres

There are two \mathbb Z_2 orbifolds of the Podleś quantum two-sphere, one being the quantum two-disc D_q and other the quantum two-dimensional real projective space \mathbb RP^2_q . In this article we calculate the Hochschild and cyclic homology and cohomology groups of these orbifolds and also the corresponding Chern–Connes indices.

Compactness of the automorphism group of a topological parallelism on real projective 3-space: The disconnected case

Compactness of the automorphism group of a topological parallelism on real projective 3-space: The disconnected case

Rigidity of complex convex divisible sets

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts cocompactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have [Formula: see text] boundary, and have word hyperbolic dividing group. In this paper we study a notion of convexity in complex projective space and show that the only divisible complex convex sets with [Formula: see text] boundary are the projective balls.

The smallest Laplace eigenvalue of homogeneous 3-spheres

We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace–Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of three-dimensional Berger spheres, we obtain a full description of their spectra. We also give several consequences of the mentioned expression. One of them improves known estimates for the smallest non-zero eigenvalue in terms of the diameter for homogeneous 3-spheres. Another application shows that the spectrum of the Laplace–Beltrami operator distinguishes up to isometry any left-invariant metric on SU(2). It is also proved the non-existence of constant scalar curvature metrics conformal and arbitrarily close to any non-round homogeneous metric on the 3-sphere. All of the above results are extended to left-invariant metrics on SO(3), that is, homogeneous metrics on the three-dimensional real projective space.

Dynamics of real projective transformations

<div data-canvas-width="358.1721365118214">The dynamics of a projective transformation on a real projective space are studied in this paper. The two main aspects of these transformations that are studied here are the topological entropy and the zeta function. Topological entropy is an inherent property of a dynamical system whereas the zeta function is a useful tool for the study of periodic points. We find the zeta function for a general projective transformation but entropy only for certain transformations on the real projective line.</div>

Bounds for higher topological complexity of real projective space implied by BP

We use Brown–Peterson cohomology to obtain lower bounds for the higher topological complexity, $${\text {TC}}_k(\hbox {RP}^{2m})$$ , of real projective spaces, which are often much stronger than those implied by ordinary mod-2 cohomology.

Symmetric Bi-Skew Maps and Symmetrized Motion Planning in Projective Spaces

AbstractThis work is motivated by the question of whether there are spacesXfor which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space${\open R}\hbox{P}^m$, it is known that$\hbox{TC}^S ({\open R}\hbox{P}^{m})$captures, with a few potential exceptional cases, the Euclidean embedding dimension of${\open R}\hbox{P}^{m}$. We now show that, for allm≥1,$\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$is characterized as the smallest positive integernfor which there is a symmetric${\open Z}_{2}$-biequivariant mapSm×Sm→Snwith a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of${\open R}\hbox{P}^{2^{e}}$fore≥1. In particular, this leaves the torusS1×S1as the only closed surface whose symmetric (symmetrized) TCS(TCΣ) invariant is currently unknown.