arXiv research

We classify homotopes of classical symmetric spaces (studied in Part I of this work). Our classification uses the fibered structure of homotopes: they are fibered as symmetric spaces, with flat fibers, over a non-degenerate base; the base spaces correspond to inner ideals in Jordan pairs. Using that inner ideals in cla…

We prove that the tangent bundle of an inner symmetric space $M$ of compact type is weakly complex if and only if $M$ is a Riemannian product $M_1\times...\times M_k$, each $M_i$ being an even-dimensional round sphere or Hermitian symmetric.

In this paper, we develop a loop group description of harmonic maps $\mathcal{F}: M \rightarrow G/K$ ``of finite uniton type", from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type. This develops work of Uhlenbeck, Segal, and Burstall-Guest to non-compact inner symmetric spaces. To be mo…

We prove that compact quaternionic-Kähler manifolds of positive scalar curvature admit no almost complex structure, even in the weak sense, except for the complex Grassmannians $Gr_2(C^{n+2})$. We also prove that irreducible inner symmetric spaces $M^{4n}$ of compact type are not weakly complex, except for spheres and …

Loop group method varies with base point choice.

We find a one-to-one correspondence between full extrinsic symmetric spaces in (possibly degenerate) inner product spaces and certain algebraic objects called (weak) extrinsic symmetric triples. In particular, this yields a description of arbitrary extrinsic symmetric spaces in pseudo-Euclidean spaces by corresponding …

Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.

Paper develops a new method for harmonic maps into symmetric spaces.

In this note, we show that for any harmonic map into a non-compact symmetric space one can find naturally a "dual" harmonic map into a compact symmetric space which can be constructed from the same basic data (called "potentials" in the loop group formalism). Locally also the inverse/converse duality theorem holds.

In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit exp…

New infinite families of flat spaces found from symmetric spaces.

This study approximates neural network features for modeling relations and attention mechanisms.

Let G/H be a compact 4-symmetric space of inner type such that the dimension of the center Z(H) of H is at most one. In this paper we shall classify involutions of G preserving H for the case where dim Z(H)=0, or H is a centralizer of a toral subgroup of G.

The paper explores harmonic maps and their properties in symmetric spaces.

This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.

Let $G/K$ be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of $G$ sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of $G$. As an example of application, we give the list of the …

We describe complex twistor spaces over inner 3-symmetric spaces $G/H$, such that $H$ acts transitively on the fibre. Like in the symmetric case, these are flag manifolds $G/K$ where $K$ is the centralizer of a torus in $G$. Moreover, they carry an almost complex structure defined using the horizontal distribution of t…

Survey of Willmore surfaces in spheres using DPW method.

In this paper we deal with the global properties of Willmore surfaces in spheres via the harmonic conformal Gauss map using loop groups. We first derive a global description of those harmonic maps which can be realized as conformal Gauss maps of some Willmore surfaces (Theorem 3.4, Theorem 3.11 and Theorem 3.18). Then …

Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.

We consider the problem of designing locality sensitive hashes (LSH) for inner product similarity, and of the power of asymmetric hashes in this context. Shrivastava and Li argue that there is no symmetric LSH for the problem and propose an asymmetric LSH based on different mappings for query and database points. Howev…

We show that given a harmonic map $\varphi$ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a $J_2$-holomorphic twistor lift of $\varphi$ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae i…

We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.

We define a quandle variety as an irreducible algebraic variety $Q$ endowed with an algebraically defined quandle operation $\rhd$. It can also be seen as an analogue of a generalized affine symmetric space or a regular $s$-manifold in algebraic geometry. Assume that $Q$ is normal as an algebraic variety and that the a…

Paper proposes a new method to optimize feature coordinates for better image classification.

Let $A$ be a positive injective operator in a Hilbert space (\h, <,>), and denote by [,] the inner product defined by A: [f,g]=<Af,g>. A closed subspace $\s \subset \h$ is called A-compatible if there exists a closed complement for $\s$, which is orthogonal to $\s$ with respect to the inner product [,]. Equivalently, i…

A graph (digraph) $G=(V,E)$ with a set $T\subseteq V$ of terminals is called inner Eulerian if each nonterminal node $v$ has even degree (resp. the numbers of edges entering and leaving $v$ are equal). Cherkassky and Lovász showed that the maximum number of pairwise edge-disjoint $T$-paths in an inner Eulerian graph $G…

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $π$ is a spacelike 2 plane, let $R(π)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hyp…

This research simplifies Riemannian LBFGS for SPD manifolds.

We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalga…

The tangent hyperplanes of the "manifolds" of this paper equipped a so-called Minkowski product. It is neither symmetric nor bilinear. We give a method to handing such an object as a locally hypersurface of a generalized space-time model and define the main tools of its differential geometry: its fundamental forms, its…

Completed realizations of automorphisms and subgroups in exceptional Lie group $E_8$.

Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces CP^n are classified, except if n=15 and q=1. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations…

We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank …

We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formu…

The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class $Ω$…

We present here a result of Monomialization of real analytic two-symmetric tensor fields over regular real analytic surfaces. We apply it to the (extension of the pull-back of the) inner metric of a resolved surface of a real analytic surface singularity. Doing so we recover Hsiang & Pati property at each point of the …

Researchers prove inner product recovery is impossible in latent space models.

Study higher rank inner products and their tilings to describe tori degenerations.

Convex learning for diverse invariances in semi-inner-product space.

The paper shows BNSR-invariants of McCool groups are either dense or empty.

The paper proves positive energy-momentum theorems for charged AdS initial data sets.

Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tildeγ$ of order four on $G$ and $H$ is a fixed point…

We study one extremal problem on the product of power of generalized inner radii of non-overlapping domains in $\mathbb{C}^{n}$.

We describe a construction of fibrewise inner products on the cotangent bundle of the smooth free loop space of a Riemannian manifold. Using this inner product, we construct an operator over the loop space of a string manifold which is directly analogous to the Dirac operator of a spin manifold.

We investigate the properties of principal elements of Frobenius Lie algebras, following the work of M. Gerstenhaber and A. Giaquinto. We prove that any Lie algebra with a left symmetric algebra structure can be embedded, in a natural way, as a subalgebra of some sl(m,K), for K= R or C. Hence, the work of Belavin and D…

We give a condition for a function to produce a Möbius invariant weighted inner product on the tangent space of the space of knots, and show that some kind of Möbius invariant knot energies can produce Möbius invariant and parametrization invariant weighted inner products. They would give a natural way to study the evo…