scholarly journals Flexibility of Embeddings of Bouquets of Circles on the Projective Plane and Klein Bottle

10.37236/998 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Yan Yang ◽  
Yanpei Liu
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Equivalence Classes ◽  
Isomorphism Classes ◽  
Nonorientable Surfaces ◽  
Explicit Expressions

In this paper, we study the flexibility of embeddings of bouquets of circles on the projective plane and the Klein bottle. The numbers (of equivalence classes) of embeddings of bouquets of circles on these two nonorientable surfaces are obtained in explicit expressions. As their applications, the numbers (of isomorphism classes) of rooted one-vertex maps on these two nonorientable surfaces are deduced.

scholarly journals Checkerboard embeddings of *-graphs into nonorientable surfaces

2014 ◽  
Vol 23 (07) ◽  
pp. 1460004
Author(s):  
Tyler Friesen ◽  
Vassily Olegovich Manturov
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Orientable Surface ◽  
Nonorientable Surfaces ◽  
Embeddability Condition

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper [Embeddings of *-graphs into 2-surfaces, preprint (2012), arXiv:1212.5646] by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admits a ℤ2-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a *-graph with all vertices of degree 4 or 6 admits a ℤ2-homologically trivial embedding into the projective plane or into the Klein bottle.

scholarly journals Classification of ($p,q,n$)-Dipoles on Nonorientable Surfaces

10.37236/461 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yan Yang ◽  
Yanpei Liu
Keyword(s):  
String Theory ◽  
Projective Plane ◽  
Klein Bottle ◽  
Electronic Journal ◽  
Nonorientable Surfaces ◽  
Rooted Map ◽  
Orientable Surfaces

A type of rooted map called $(p,q,n)$-dipole, whose numbers on surfaces have some applications in string theory, are defined and the numbers of $(p,q,n)$-dipoles on orientable surfaces of genus 1 and 2 are given by Visentin and Wieler (The Electronic Journal of Combinatorics 14 (2007),#R12). In this paper, we study the classification of $(p,q,n)$-dipoles on nonorientable surfaces and obtain the numbers of $(p,q,n)$-dipoles on the projective plane and Klein bottle.

Enumeration of unsensed r-regular maps on the projective plane and the Klein bottle

2021 ◽  
Vol 344 (11) ◽  
pp. 112528
Author(s):  
Evgeniy Krasko ◽  
Alexander Omelchenko
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Regular Maps

scholarly journals Invariant connections and invariant holomorphic bundles on homogeneous manifolds

2014 ◽  
Vol 12 (1) ◽  
pp. 1-13
Author(s):  
Indranil Biswas ◽  
Andrei Teleman
Keyword(s):  
Lie Group ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Transitive Action ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Group A ◽  
Holomorphic Bundles ◽  
Invariant Gauge

AbstractLet X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

The Enumeration of Maps on the Torus and the Projective Plane

1988 ◽  
Vol 31 (3) ◽  
pp. 257-271 ◽  
Author(s):  
E. A. Bender ◽  
E. R. Canfield ◽  
R. W. Robinson
Keyword(s):  
Projective Plane ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Embedded Graphs ◽  
Asymptotic Expressions ◽  
Explicit Expressions

AbstractThe enumeration of rooted maps (embedded graphs), by number of edges, on the torus and projective plane, is studied. Explicit expressions for the generating functions are obtained. From these are derived asymptotic expressions and recurrence relations. Numerical tables for the numbers with up to 20 edges are presented.

On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces

2006 ◽  
Vol 49 (4) ◽  
pp. 624-627
Author(s):  
Masakazu Teragaito
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Orientable Surface ◽  
Dehn Surgery ◽  
Orientable Surfaces

AbstractFor a non-trivial knot in the 3-sphere, only integral Dehn surgery can create a closed 3-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

scholarly journals Isomorphism classes for higher order tangent bundles

2017 ◽  
Vol 17 (2) ◽  
pp. 175-189 ◽  
Author(s):  
Ali Suri
Keyword(s):  
Vector Bundle ◽  
Linear Connection ◽  
Equivalence Classes ◽  
Banach Manifold ◽  
Invariant Vector ◽  
Isomorphism Classes ◽  
Connection Map ◽  
Tangent Bundles ◽  
Order Tangent ◽  
Bundle Structure

AbstractThe tangent bundle TkM of order k of a smooth Banach manifold M consists of all equivalence classes of curves that agree up to their accelerations of order k. In previous work the author proved that TkM, 1 ≤ k ≤∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection, or equivalently if a connection map on TkM is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the k-th order differential Tkg : TkM ⟶ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M ⟶ M is a diffeomorphism, one obtains invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combinations of connection maps and manifolds of Cr maps we offer three examples for our theory, showing its interaction with known problems such as the Sasaki lift of metrics.

SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF RIEMANNIAN MANIFOLDS

2004 ◽  
Vol 06 (06) ◽  
pp. 867-879 ◽  
Author(s):  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Projective Plane ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Klein Bottle ◽  
Euler Class ◽  
Isometric Immersions ◽  
Existence And Nonexistence

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP2 is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S3(1), solving a question posed by Gromov.

scholarly journals REDUCING DEHN FILLINGS AND SMALL SURFACES

2005 ◽  
Vol 92 (1) ◽  
pp. 203-223 ◽  
Author(s):  
SANGYOP LEE ◽  
SEUNGSANG OH ◽  
MASAKAZU TERAGAITO
Keyword(s):  
Projective Plane ◽  
Klein Bottle ◽  
Hyperbolic Manifolds ◽  
The Other ◽  
Möbius Band ◽  
Dehn Fillings ◽  
Orientable Surfaces

In this paper we investigate the distances between Dehn fillings on a hyperbolic 3-manifold that yield 3-manifolds containing essential small surfaces including non-orientable surfaces. In particular, we study the situations where one filling creates an essential sphere or projective plane, and the other creates an essential sphere, projective plane, annulus, Möbius band, torus or Klein bottle, for all eleven pairs of such non-hyperbolic manifolds.

scholarly journals Distance-Restricted Matching Extension in Triangulations of the Torus and the Klein Bottle

10.37236/2952 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Robert E.L. Aldred ◽  
Jun Fujisawa
Keyword(s):  
Projective Plane ◽  
Perfect Matching ◽  
Klein Bottle ◽  
Even Order

A graph $G$ with at least $2m+2$ edges is said to be distance $d$ $m$-extendable if for any matching $M$ in $G$ with $m$ edges in which the edges lie pair-wise distance at least $d$, there exists a perfect matching in $G$ containing $M$. In a previous paper, Aldred and Plummer proved that every $5$-connected triangulation of the plane or the projective plane of even order is distance $5$ $m$-extendable for any $m$. In this paper we prove that the same conclusion holds for every triangulation of the torus or the Klein bottle.

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