Abstract We give a sharp lower bound for energy in homotopy classes of maps from real projective space to Riemannian manifolds, together with an upper bound for the infimum of the energy in such a homotopy class. We characterize the maps attaining this lower bound for energy, and we explain how the infimum of the energy in a homotopy class of maps of real projective ‐space is determined by an associated class of maps of the real projective plane.

A result of Teissier says that the cone over one of classical polygon examples in the real projective space gives, by complexification, a surface singularity which is not Whitney equisingular to a singularity defined over the field Q of rational numbers. In this note we correct the example and give a complete proof of Teissier’s result.

In this paper, we prove an upper bound on the second nonzero Laplacian eigenvalue on n -dimensional real projective space. The sharp result for 2-dimensions was shown by Nadirashvili and Penskoi and later by Karpukhin when the metric degenerates to that of the disjoint union of a round projective space and a sphere. That conjecture is open in higher dimensions, but this paper proves it up to a constant factor that tends to 1 as the dimension tends to infinity. Also, we introduce a topological argument that deals with the orthogonality conditions in a single step proof.

Abstract We study the space of paths in a closed manifold M with endpoints determined by an involution $$f:M\rightarrow M$$ f : M → M . If the involution is fixed point free and if M is 2-connected then this path space is the universal covering space of the component of non-contractible loops of the free loop space of $$M/\mathbb {Z}_2$$ M / Z 2 . On the homology of said path space we study string topology operations which extend the Chas–Sullivan loop product and the Goresky–Hingston loop coproduct, respectively. We study the case of antipodal involution on the sphere in detail and use Morse–Bott theoretic methods to give a complete computation of the extended loop product and the extended coproduct on even-dimensional spheres. These results are then applied to prove a resonance theorem for closed geodesics on real projective space.

Recent advancements in quantum polarization theory have propelled the exploration of topological insulators (TIs) into the realm of higher-order systems, leading to the study of the celebrated two-dimensional (2D) quadrupole and 3D octupole TIs. Traditionally, these topological phases have been associated with the toroidal topology of the conventional Brillouin zone. This paper reports the discovery of a novel octupole topological insulating phase protected by a 3D momentum-space nonsymmorphic group emerging within the framework of the Brillouin 3D real projective space ([Formula: see text]). We theoretically propose the model and its corresponding topological invariant, experimentally construct this insulator within a topological circuit framework and capture the octupole insulating phase as a localized impedance peak at the circuit's corner. Furthermore, our [Formula: see text] circuit stands out as a pioneering 3D model to simultaneously exhibit both intrinsic, termination-independent symmetry-protected topological phases and extrinsic, termination-dependent surface-obstructed topological phases within the symmetry-protected topological phases. Our results broaden the topological landscape and provide insights into the band theory within the manifold of the Brillouin [Formula: see text] space.

Abstract In the previous paper (Skriganov, J. Complexity 56 (2020), 101428), Stolarsky's invariance principle, known in the literature for point distributions on Euclidean spheres, has been extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. Geometric features of these spaces as well as their models in terms of Jordan algebras have been used very essentially in the proof. In the present paper, a new pure analytic proof of the extended Stolarsky's invariance principle is given, relying on the theory of spherical functions on compact Riemannian symmetric manifolds of rank one.

Strictly positive definite functions on spheres

A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space Rn with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode the orientations of hyperplanes and spheres. The natural space for encoding such geometric objects is the real projective quadric with signature (n+1,2). In this article, we first provide a general formula for calculating the angles formed by the geometric objects encoded by the points of the quadric. The main result is the determination of a very simple parametrization of a Laguerre-type subgroup that acts transitively on the quadric while preserving the geometric nature of its points. That is, points of the quadric representing oriented spheres, points, and oriented hyperplanes in Rn are mapped by the action to points of the same geometric type. We also provide simple parametrizations of the isotropies of the action. The action described in this article provides the foundation for an effective solution to geometric constraint problems.

We classify totally geodesic submanifolds in Hopf–Berger spheres, which constitute a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations. As a byproduct of our investigations, we have discovered very intriguing examples of totally geodesic submanifolds: totally geodesic submanifolds isometric to real projective spaces, uncountably many isometric but non-congruent totally geodesic submanifolds, and a totally geodesic submanifold that is not extrinsically homogeneous. Remarkably, all these examples arise in certain Hopf–Berger spheres with positive curvature.

Abstract We study asymptotics of the eigenvalues and eigenfunctions of the operators used for constructing multidimensional scaling (MDS) on closed connected symmetric spaces. They are the limits of eigenvalues and eigenvectors of squared distance matrices of an increasing sequence of finite subsets covering the space densely in the limit. We show that for products of spheres and real projective spaces, the numbers of positive and negative eigenvalues of these operators are both infinite. We also find a class of spaces (namely $\mathbb{RP}^{n}$ with odd $n>1$) whose MDS defining operators are not trace class, and original distances cannot be reconstructed from the eigenvalues and eigenfunctions of these operators.

ABSTRACTIn this contribution, we consider the optimal shape design of thin elastic shell structures based on a linearized shell model of Koiter's type, whose shape can be described by a surface immersed in three‐dimensional Euclidean space. We regard the set of unparametrized immersions of the surface as an infinite‐dimensional Riemannian shape space and perform optimization in this setting using the Riemannian shape gradient. Nonuniform rational basis splines (NURBS) are employed to discretize the shell and numerically solve the underlying equations that govern its mechanical behavior via isogeometric analysis. By representing NURBS patches as B‐spline patches in real projective space, NURBS weights can also be incorporated into the optimization routine. We discuss the practical implementation of the method and demonstrate our approach on the compliance minimization of a half‐cylindrical shell under static load and fixed area constraint.

Abstract This paper discusses the development of synthetic cohomology in Homotopy Type Theory (HoTT), as well as its computer formalisation. The objectives of this paper are (1) to generalise previous work on integral cohomology in HoTT by the current authors and Brunerie (2022) to cohomology with arbitrary coefficients and (2) to provide the mathematical details of, as well as extend, results underpinning the computer formalisation of cohomology rings by the current authors and Lamiaux (2023). With respect to objective (1), we provide new direct definitions of the cohomology group operations and of the cup product, which, just as in the previous work by the current authors and Brunerie (2022), enable significant simplifications of many earlier proofs in synthetic cohomology theory. In particular, the new definition of the cup product allows us to give the first complete formalisation of the axioms needed to turn the cohomology groups into a graded commutative ring. We also establish that this cohomology theory satisfies the HoTT formulation of the Eilenberg–Steenrod axioms for cohomology and study the classical Mayer–Vietoris and Gysin sequences. With respect to objective (2), we characterise the cohomology groups and rings of various spaces, including the spheres, torus, Klein bottle, real/complex projective planes, and infinite real projective space. All results have been formalised in Cubical Agda, and we obtain multiple new numbers, similar to the famous ‘Brunerie number’, which can be used as benchmarks for computational implementations of HoTT. Some of these numbers are infeasible to compute in Cubical Agda and hence provide new computational challenges and open problems which are much easier to define than the original Brunerie number.

In [Ye et al., Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. In this article, we make an interdisciplinary global study of Class I. Since the family has four parameters, we have studied it using the same technique that has already been used in several papers with similar systems which is based on the algebraic invariants of the Sibirskii's school. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 261 subsets with 49 different phase portraits for Class I (2 of them corresponding to linear systems), 7 of which have limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by an analytic set of curves corresponding to phase portraits which have separatrix connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections.

We consider type IIB supergravity on a Z_{2} quotient of AdS_{5}×S^{5} as the holographic dual of strongly coupled 4D N=4 SYM on RP^{4} space with the gauging of charge conjugation. Using bootstrap techniques, we determine all two-point functions of 1/2-BPS operators of arbitrary weights at the leading order in the large central charge expansion.

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

Abstract In 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.

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.

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

On multiplicative spectral sequences for nerves and the free loop spaces