scholarly journals BOUNDS ON THE DIMENSION OF MANIFOLDS WITH INVOLUTION FIXING Fn ∪ F2

2008 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
PEDRO L. Q. PERGHER ◽  
FÁBIO G. FIGUEIRA
Keyword(s):  
Fixed Point ◽  
Upper Bound ◽  
Normal Bundle ◽  
Smooth Manifold ◽  
Upper Bounds ◽  
Fixed Point Set ◽  
Similar Question ◽  
Cobordism Class ◽  
Point Set

AbstractLet Mm be a closed smooth manifold with an involution having fixed point set of the form Fn ∪ F2, where Fn and F2 are submanifolds with dimensions n and 2, respectively, where n ≥ 4 is even (n < m). Suppose that the normal bundle of F2 in Mm, μ → F2, does not bound, and denote by β the stable cobordism class of μ → F2. In this paper, we determine the upper bound for m in terms of the pair (n, β) for many such pairs. The similar question for n odd (n ≥ 3) was completely solved in a previous paper of the authors. The existence of these upper bounds is guaranteed by the famous 5/2-theorem of Boardman, which establishes that, under the above hypotheses, m ≤ 5/2n.

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

Rational Integer Invariants of Regular Cyclic Actions

2004 ◽  
Vol 47 (1) ◽  
pp. 60-72 ◽  
Author(s):  
Robert D. Little
Keyword(s):  
Fixed Point ◽  
Normal Bundle ◽  
Complex Structure ◽  
Rational Integer ◽  
Fixed Point Set ◽  
Cyclic Action ◽  
Point Set ◽  
Smooth Map

AbstractLet g : M2n → M2n be a smooth map of period m ≥ 2 which preserves orientation. Suppose that the cyclic action defined by g is regular and that the normal bundle of the fixed point set F has a g-equivariant complex structure. Let F ⋔ F be the transverse self-intersection of F with itself. If the g-signature Sign(g, M) is a rational integer and n < ϕ(m), then there exists a choice of orientations such that Sign(g, M) = Sign F = Sign(F ⋔ F).

scholarly journals Tangent Dirac Structures and Poisson Dirac Submanifolds

2015 ◽  
Vol 62 (1) ◽  
pp. 21-24
Author(s):  
Md Showkat Ali ◽  
MG M Talukder ◽  
MR Khan
Keyword(s):  
Fixed Point ◽  
Poisson Structure ◽  
Normal Bundle ◽  
Dirac Structure ◽  
Fixed Point Set ◽  
Dirac Structures ◽  
Coisotropic Submanifold ◽  
Point Set

The local equations that characterize the submanifolds N of a Dirac manifold M is an isotropic (coisotropic) submanifold of TM endowed with the tangent Dirac structure. In the Poisson case which is a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure if and only if N is a Dirac submanifold. In this paper we have proved a theorem in the general Poisson case that the fixed point set MG has a natural induced Poisson structure that implies a Poisson-Dirac submanifolds, where G×M?M be a proper Poisson action. DOI: http://dx.doi.org/10.3329/dujs.v62i1.21955 Dhaka Univ. J. Sci. 62(1): 21-24, 2014 (January)

scholarly journals Involutions Fixing Fn ∪ ﹛Indecomposable﹜

2012 ◽  
Vol 55 (1) ◽  
pp. 164-171 ◽  
Author(s):  
Pedro L. Q. Pergher
Keyword(s):  
Fixed Point ◽  
Smooth Manifold ◽  
Fixed Point Set ◽  
Point Set

AbstractLet Mm be an m-dimensional, closed and smooth manifold, equipped with a smooth involution T : Mm → Mm whose fixed point set has the form Fn ∪ Fj, where Fn and Fj are submanifolds with dimensions n and j, Fj is indecomposable and n > j. Write n – j = 2pq, where q ≥ 1 is odd and p ≥ 0, and set m(n– j) = 2n + p–q +1 if p ≤ q +1 and m(n– j) = 2n +2p–q if p ≥ q. In this paper we show that m ≤ m(n – j) + 2j + 1. Further, we show that this bound is almost best possible, by exhibiting examples (Mm(n–j)+2j, T) where the fixed point set of T has the form Fn ∪ Fj described above, for every 2 ≤ j < n and j not of the form 2t – 1 (for j = 0 and 2, it has been previously shown that m(n – j) + 2 j is the best possible bound). The existence of these bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that under the above hypotheses .

scholarly journals Involutions whose fixed set has three or four components: a small codimension phenomenon

2012 ◽  
Vol 110 (2) ◽  
pp. 223 ◽  
Author(s):  
Evelin M. Barbaresco ◽  
Patricia E. Desideri ◽  
Pedro L. Q. Pergher
Keyword(s):  
Fixed Point ◽  
Smooth Manifold ◽  
Fixed Point Set ◽  
Normal Bundles ◽  
Fixed Set ◽  
Point Set ◽  
Small Codimension

Let $T:M \to M$ be a smooth involution on a closed smooth manifold and $F = \bigcup_{j=0}^n F^j$ the fixed point set of $T$, where $F^j$ denotes the union of those components of $F$ having dimension $j$ and thus $n$ is the dimension of the component of $F$ of largest dimension. In this paper we prove the following result, which characterizes a small codimension phenomenon: suppose that $n \ge 4$ is even and $F$ has one of the following forms: 1) $F=F^n \cup F^3 \cup F^2 \cup \{{\operatorname {point}}\}$; 2) $F=F^n \cup F^3 \cup F^2 $; 3) $F=F^n \cup F^3 \cup \{{\operatorname{point}}\}$; or 4) $F=F^n \cup F^3$. Also, suppose that the normal bundles of $F^n$, $F^3$ and $F^2$ in $M$ do not bound. If $k$ denote the codimension of $F^n$, then $k \le 4$. Further, we construct involutions showing that this bound is best possible in the cases 2) and 4), and in the cases 1) and 3) when $n$ is of the form $n=4t$, with $t \ge 1$.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

scholarly journals On soft quasi-pseudometric spaces

2021 ◽  
Vol 22 (1) ◽  
pp. 17
Author(s):  
Hope Sabao ◽  
Olivier Olela Otafudu
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Soft Set ◽  
Fixed Point Set ◽  
Point Set ◽  
The Absolute ◽  
Pseudometric Space

<p>In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set.</p>

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