arXiv research

A new retraction on Stiefel manifold with a closed-form inverse.

This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.

New retraction on symplectic Stiefel manifold with closed-form inverse.

Study numerical invariants under retraction maps between topological spaces.

NR retraction approximates geodesics on submanifolds efficiently.

Contact group retracts to unitary subgroup.

Topological manifolds can be embedded flatly in high-dimensional Euclidean space and are locally retracts.

In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation retracts, of the Teichmüller space $\T_g$ of compact Riemann surfaces of genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive codimension and construct an intrinsic $\Mod_g$-equivariant deformation retraction from $\T_g$…

A new algorithm avoids retractions to optimize orthogonal matrices efficiently.

Study random walks on sub-Riemannian manifolds using retractions.

The present research work proposes a new fast fixed-point averaging algorithm on the compact Stiefel manifold based on a mixed retraction/lifting pair. Numerical comparisons between fixed-point algorithms based on the proposed non-associated retraction/lifting map pair and two associated retraction/lifting pairs confir…

Construct special Lagrangian fibrations on abelian varieties using retraction techniques.

Embeds complex into higher-dimensional pseudomanifold.

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

We prove that the well-rounded retract of SO_n\SL_n(R) is a minimal SL_n(Z)-invariant spine.

Modernizes classical theory linking isothermic surfaces to Bonnet pairs.

Constructs harmonic maps near retractions in hyperbolic spaces.

The paper creates a deformation retraction for homeomorphisms of the projective plane.

The paper compares numerical schemes for nonholonomic systems using retraction maps.

We characterize metric spaces $X$ whose hyperspaces $2^X$ or $Bd(X)$ of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.

Simplicial sets deformation retract onto transverse simplices.

This note surveys axiomatic results for the Farrell-Jones Conjecture in terms of actions on Euclidean retracts and applications of these to GL_n(Z), relative hyperbolic groups and mapping class groups.

Outer space and Teichmüller space fail well-rounded retract analogy.

This research solves Hermite interpolation on manifolds using retractions.

Smale proved that the orientation-preserving diffeomorphism group of S^2 has a continuous strong deformation retraction to SO(3). In this paper, we construct such a strong deformation retraction which is diffeologically smooth.

Polyhedra collapse to subpolyhedra if they can be continuously shrunk onto them.

We show that the nearest point retraction is a uniform quasi-isometry from the Thurston metric on a hyperbolic domain in the Riemann sphere to the boundary of the convex hull of its complement. As a corollary, one obtains explicit bounds on the quasi-isometry constant of the nearest point retraction with respect to the…

Being a maximal compact subgroup of SL_nC, SU_n is a deformation retract of the former group. In this note we prove that, for sufficiently large n, there is no retraction of SL_nC to SU_n which preserves commutativity.

Global homotopies upgrade classical map in differential geometry.

Spaces of circle embeddings in curved surfaces indexed by trees.

Schmutz Schaller and Thurston's approaches are dual.

We obtain a classification up to isomorphism of complex-analytic supermanifolds with underlying space $\mathbb{CP}^1$ of dimension $1|3$ with retract $(k,k,k)$, where $k\in \mathbb{Z}$. More precisely, we prove that classes of isomorphic complex-analytic supermanifolds of dimension $1|3$ with retract $(k,k,k)$ are in o…

We construct a one-dimensional deformation retract of the unordered k-point configuration space of a star S. This retract suggests an explicit set of free generators Beta_k for the corresponding braid group of the star B_k and shows that the natural map from B_k-1 to B_k sends Beta_k-1 to Beta_k injectively.

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

New integrators for mechanical systems on Lie groups simplify based on group properties.

Let $Γ=\langle a,b | a b^{p} a^{-1} = b^{q}\rangle$ be a Baumslag--Solitar group and $G$ be a complex reductive algebraic group with maximal compact subgroup $K<G$. We show that, when $p$ and $q$ are relatively prime with distinct absolute values, there is a strong deformation retraction retraction of $Hom(Γ,G)$ onto $…

We show that the arc graph of $S_g^1$ is a coarse Lipschitz retract of the free splitting complex of $F_{2g}$. We also show that the arc and curve graph of $S_g^1$ is a coarse Lipschitz retract of both the cyclic splitting graph of $F_{2g}$ and the maximally cyclic splitting graph of $F_{2g}$.

Study on higher-dimensional black holes, focusing on retractions and scalar quasibound states.

We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first ty…

Control data constructed for smooth weak deformation retraction of stratified spaces.

Newton's method tackles nonlinear mappings into vector bundles with connections and retractions.

Stochastic variance reduction algorithms have recently become popular for minimizing the average of a large, but finite number of loss functions. The present paper proposes a Riemannian stochastic quasi-Newton algorithm with variance reduction (R-SQN-VR). The key challenges of averaging, adding, and subtracting multipl…

Retracted temporarily to fix some erratic things.

Let S be a compact surface, and M be the double of a handlebody. Given a homotopy class of maps from S to M inducing an isomorphism of fundamental groups, we describe a canonical uniformly lipschitz retraction of the sphere graph of M to the arc graph of S. We also show that this retraction is a uniformly bounded dista…

We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall. Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an …

Given a hyperbolic domain, the nearest point retraction is a conformally natural homotopy equivalence from the domain to the boundary of the convex core of its complement. Marden and Markovic showed that if the domain is uniformly perfect, then there exists a conformally natural quasiconformal map which admits a bounde…

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …

An important part of the classical theory of real or complex manifolds is the theory of (smooth, real analytic or complex analytic) vector bundles. With any vector bundle over a manifold (M,F) the sheaf of its (smooth, real analytic or complex analytic) sections is associated which is a locally free sheaf of F-modules,…