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Artigos de revistas sobre o tema "Surfaces non orientables"

Autor: Grafiati

Publicado: 19 de outubro de 2024

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1

Oliveira, M. Elisa G. G., e Eric Toubiana. "Surfaces non-orientables de genre deux". Boletim da Sociedade Brasileira de Matem�tica 24, n.º 1 (março de 1993): 63–88. http://dx.doi.org/10.1007/bf01231696.

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2

Toubiana, E. "Surfaces minimales non orientables de genre quelconque". Bulletin de la Société mathématique de France 121, n.º 2 (1993): 183–95. http://dx.doi.org/10.24033/bsmf.2206.

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3

Bhowmik, Debashis, Dipendu Maity e Eduardo Brandani Da Silva. "Surface codes and color codes associated with non-orientable surfaces". Quantum Information and Computation 21, n.º 13&14 (setembro de 2021): 1135–53. http://dx.doi.org/10.26421/qic21.13-14-4.

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Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus $\ge 1$ and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus $ \geq 2n+1 $ for $n\geq 1$. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios $k/n$ and $d/n$ better than the codes obtained from orientable surfaces.

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4

NAKAMURA, GOU. "COMPACT NON-ORIENTABLE SURFACES OF GENUS 5 WITH EXTREMAL METRIC DISCS". Glasgow Mathematical Journal 54, n.º 2 (12 de dezembro de 2011): 273–81. http://dx.doi.org/10.1017/s0017089511000589.

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AbstractA compact hyperbolic surface of genus g is called an extremal surface if it admits an extremal disc, a disc of the largest radius determined by g. Our problem is to find how many extremal discs are embedded in non-orientable extremal surfaces. It is known that non-orientable extremal surfaces of genus g > 6 contain exactly one extremal disc and that of genus 3 or 4 contain at most two. In the present paper we shall give all the non-orientable extremal surfaces of genus 5, and find the locations of all extremal discs in those surfaces. As a consequence, non-orientable extremal surfaces of genus 5 contain at most two extremal discs.

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5

Yurttaş, S. Öykü. "Curves on Non-Orientable Surfaces and Crosscap Transpositions". Mathematics 10, n.º 9 (28 de abril de 2022): 1476. http://dx.doi.org/10.3390/math10091476.

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Let Ng,n be a non-orientable surface of genus g with n punctures and one boundary component. In this paper, we describe multicurves in Ng,n making use of generalized Dynnikov coordinates, and give explicit formulae for the action of crosscap transpositions and their inverses on the set of multicurves in Ng,n in terms of generalized Dynnikov coordinates. This provides a way to solve on non-orientable surfaces various dynamical and combinatorial problems that arise in the study of mapping class groups and Thurston’s theory of surface homeomorphisms, which were solved only on orientable surfaces before.

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6

Sabloff, Joshua. "On a refinement of the non-orientable 4-genus of Torus knots". Proceedings of the American Mathematical Society, Series B 10, n.º 22 (21 de junho de 2023): 242–51. http://dx.doi.org/10.1090/bproc/166.

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In formulating a non-orientable analogue of the Milnor Conjecture on the 4 4 -genus of torus knots, Batson [Math. Res. Lett. 21 (2014), pp. 423–436] developed an elegant construction that produces a smooth non-orientable spanning surface in B 4 B^4 for a given torus knot in S 3 S^3 . While Lobb [Math. Res. Lett. 26 (2019), pp. 1789] showed that Batson’s surfaces do not always minimize the non-orientable 4 4 -genus, we prove that they do minimize among surfaces that share their normal Euler number. We also determine the possible pairs of normal Euler number and first Betti number for non-orientable surfaces whose boundary lies in a class of torus knots for which Batson’s surfaces are non-orientable 4 4 -genus minimizers.

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7

Gastesi, Pablo Arés. "Some results on Teichmüller spaces of Klein surfaces". Glasgow Mathematical Journal 39, n.º 1 (janeiro de 1997): 65–76. http://dx.doi.org/10.1017/s001708950003192x.

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The deformation theory of nonorientable surfaces deals with the problem of studying parameter spaces for the different dianalytic structures that a surface can have. It is an extension of the classical theory of Teichmüller spaces of Riemann surfaces, and as such, it is quite rich. In this paper we study some basic properties of the Teichmüller spaces of non-orientable surfaces, whose parallels in the orientable situation are well known. More precisely, we prove an uniformization theorem, similar to the case of Riemann surfaces, which shows that a non-orientable compact surface can be represented as the quotient of a simply connected domain of the Riemann sphere, by a discrete group of Möbius and anti-Möbius transformation (mappings whose conjugates are Mobius transformations). This uniformization result allows us to give explicit examples of Teichmüller spaces of non-orientable surfaces, as subsets of deformation spaces of orientable surfaces. We also prove two isomorphism theorems: in the first place, we show that the Teichmüller spaces of surfaces of different topological type are not, in general, equivalent. We then show that, if the topological type is preserved, but the signature changes, then the deformations spaces are isomorphic. These are generalizations of the Patterson and Bers-Greenberg theorems for Teichmüller spaces of Riemann surfaces, respectively.

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8

Nowik, Tahl. "Immersions of non-orientable surfaces". Topology and its Applications 154, n.º 9 (maio de 2007): 1881–93. http://dx.doi.org/10.1016/j.topol.2007.02.007.

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9

Maloney, Alexander, e Simon F. Ross. "Holography on non-orientable surfaces". Classical and Quantum Gravity 33, n.º 18 (22 de agosto de 2016): 185006. http://dx.doi.org/10.1088/0264-9381/33/18/185006.

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10

Goulden, Ian P., Jin Ho Kwak e Jaeun Lee. "Enumerating branched orientable surface coverings over a non-orientable surface". Discrete Mathematics 303, n.º 1-3 (novembro de 2005): 42–55. http://dx.doi.org/10.1016/j.disc.2003.10.030.

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11

Singerman, David. "Orientable and non-orientable Klein surfaces with maximal symmetry". Glasgow Mathematical Journal 26, n.º 1 (janeiro de 1985): 31–34. http://dx.doi.org/10.1017/s0017089500005747.

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Let X be a bordered Klein surface, by which we mean a Klein surface with non-empty boundary. X is characterized topologically by its orientability, the number k of its boundary components and the genus p of the closed surface obtained by filling in all the holes. The algebraic genus g of X is defined by.If g≥2 it is known that if G is a group of automorphisms of X then |G|≤12(g-l) and that the upper bound is attained for infinitely many values of g ([4], [5]). A bordered Klein surface for which this upper bound is attained is said to have maximal symmetry. A group of 12(g-l) automorphisms of a bordered Klein surface of algebraic genus g is called an M*-group and it is known that a finite group G is an M*-group if and only if it is generated by 3 non-trivial elements T1, T2, T3 which obey the relations([4]).

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12

Kamada, Seiichi. "Orientable surfaces in the 4-sphere associated with non-orientable knotted surfaces". Mathematical Proceedings of the Cambridge Philosophical Society 108, n.º 2 (setembro de 1990): 299–306. http://dx.doi.org/10.1017/s0305004100069152.

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Let F be a closed connected and non-orientable surface smoothly embedded in the 4-sphere S4 with normal Euler number e(F) = 0. We note that if e(F) = 0, then the non-orientable genus n is even (ef. [7]) and the tubular neighbourhood N(F) of F in S4 which is a D2-bundle over F has a trivial I-subbundle. Let τ be a trivial I-subbundle of N(F) and let τ* = F × I ⊂ N(F) be its orthogonal I-subbundle which is twisted. Then is a closed connected genus n – 1 orientable surface smoothly embedded in S4 and doubly covers F. We call this surface a doubled surface of F in S4 (associated with τ). If a trivial I-subbundle τ is given, then we see that the knot type of F* ⊂ S4 is uniquely determined.

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13

Zwart, Gysbert. "Matrix theory on non-orientable surfaces". Physics Letters B 429, n.º 1-2 (junho de 1998): 27–34. http://dx.doi.org/10.1016/s0370-2693(98)00420-1.

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14

Cámpora, Daniel, Jaime de la Torre, Juan Carlos García Vázquez e Fernando Sancho Caparrini. "BML model on non-orientable surfaces". Physica A: Statistical Mechanics and its Applications 389, n.º 16 (agosto de 2010): 3290–98. http://dx.doi.org/10.1016/j.physa.2010.03.037.

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15

Matignon, D., e N. Sayari. "Non-Orientable Surfaces and Dehn Surgeries". Canadian Journal of Mathematics 56, n.º 5 (1 de outubro de 2004): 1022–33. http://dx.doi.org/10.4153/cjm-2004-046-9.

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AbstractLet K be a knot in S3. This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface . We look at the slope p/q of the surgery, the Euler characteristic χ() of the surface and the intersection number s between and the core of the Dehn surgery. We prove that if χ() ≥ 15 – 3q, then s = 1. Furthermore, if s = 1 then q ≤ 4 – 3χ() or K is cabled and q ≤ 8 – 5χ(). As consequence, if K is hyperbolic and χ() = –1, then q ≤ 7.

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16

Balacheff, Florent, e Daniel Massart. "Stable norms of non-orientable surfaces". Annales de l’institut Fourier 58, n.º 4 (2008): 1337–69. http://dx.doi.org/10.5802/aif.2386.

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17

Etayo Gordejuela, J. J. "Non-orientable automorphisms of Riemann surfaces". Archiv der Mathematik 45, n.º 4 (outubro de 1985): 374–84. http://dx.doi.org/10.1007/bf01198242.

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18

Conder, Marston, e Brent Everitt. "Regular maps on non-orientable surfaces". Geometriae Dedicata 56, n.º 2 (julho de 1995): 209–19. http://dx.doi.org/10.1007/bf01267644.

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19

End, Werner. "Non-orientable surfaces in 3-manifolds". Archiv der Mathematik 59, n.º 2 (agosto de 1992): 173–85. http://dx.doi.org/10.1007/bf01190680.

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20

Abbott, Steve, David M. Jackson e Terry I. Visentin. "An Atlas of Smaller Maps in Orientable and Non-Orientable Surfaces". Mathematical Gazette 85, n.º 504 (novembro de 2001): 573. http://dx.doi.org/10.2307/3621828.

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21

PULCINI, GABRIELE. "Rewriting systems for the surface classification theorem". Mathematical Structures in Computer Science 20, n.º 4 (27 de maio de 2010): 577–88. http://dx.doi.org/10.1017/s0960129510000101.

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The work reported in this paper refers to Massey's proof of the surface classification theorem based on the standard word-rewriting treatment of surfaces. We arrange this approach into a formal rewriting systemand provide a new version of Massey's argument. Moreover, we study the computational properties of two subsystems of:orfor dealing with words denoting orientable surfaces andnorfor dealing with words denoting non-orientable surfaces. We show how such properties induce an alternative proof for the surface classification in which the basic homeomorphism between the connected sum of three projective planes and the connected sum of a torus with a projective plane is not required.

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22

MA, XIANG, e PENG WANG. "COMPLETE STATIONARY SURFACES IN ${\mathbb R}^4_1$ WITH TOTAL GAUSSIAN CURVATURE – ∫ KdM = 4π". International Journal of Mathematics 24, n.º 11 (outubro de 2013): 1350088. http://dx.doi.org/10.1142/s0129167x13500882.

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We classify complete, algebraic, spacelike stationary (i.e. zero mean curvature) surfaces in four-dimensional Lorentz space [Formula: see text] with total Gaussian curvature – ∫ K d M = 4π. Such surfaces must be orientable surfaces, congruent to either the generalized catenoids or the generalized Enneper surfaces. The least total Gaussian curvature of a non-orientable algebraic stationary surface is 6π, which can be realized by Meeks' Möbius strip and its deformations (and also by a new class of non-algebraic examples). When the genus of its oriented double covering [Formula: see text] is g, we obtain the lower bound 2(g + 3)π, which is conjectured to be the best lower bound for each g.

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23

Bae, Yongju, J. Scott Carter, Seonmi Choi e Sera Kim. "Non-orientable surfaces in 4-dimensional space". Journal of Knot Theory and Its Ramifications 23, n.º 11 (outubro de 2014): 1430002. http://dx.doi.org/10.1142/s021821651430002x.

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This paper is a survey paper that gives detailed constructions and illustrations of some of the standard examples of non-orientable surfaces that are embedded and immersed in 4-dimensional space. The illustrations depend upon their 3-dimensional projections, and indeed the illustrations here depend upon a further projection into the plane of the page. The concepts used to develop the illustrations will be developed herein.

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24

Iwakura, Miwa, e Chuichiro Hayashi. "Non-orientable fundamental surfaces in lens spaces". Topology and its Applications 156, n.º 10 (junho de 2009): 1753–66. http://dx.doi.org/10.1016/j.topol.2009.03.002.

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25

Bernardi, Olivier, e Guillaume Chapuy. "Counting unicellular maps on non-orientable surfaces". Advances in Applied Mathematics 47, n.º 2 (agosto de 2011): 259–75. http://dx.doi.org/10.1016/j.aam.2010.09.001.

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26

Dranishnikov, Alexander. "On topological complexity of non-orientable surfaces". Topology and its Applications 232 (dezembro de 2017): 61–69. http://dx.doi.org/10.1016/j.topol.2017.09.022.

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27

Qiu, Weiyang. "Non-orientable Lagrangian Surfaces with Controlled Area". Mathematical Research Letters 8, n.º 6 (2001): 693–701. http://dx.doi.org/10.4310/mrl.2001.v8.n6.a1.

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28

Cimasoni, David. "Dimers on Graphs in Non-Orientable Surfaces". Letters in Mathematical Physics 87, n.º 1-2 (29 de janeiro de 2009): 149–79. http://dx.doi.org/10.1007/s11005-009-0299-2.

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29

Burton, Benjamin A., Arnaud de Mesmay e Uli Wagner. "Finding Non-orientable Surfaces in 3-Manifolds". Discrete & Computational Geometry 58, n.º 4 (9 de junho de 2017): 871–88. http://dx.doi.org/10.1007/s00454-017-9900-0.

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30

Dupont, Grégoire, e Frédéric Palesi. "Quasi-cluster algebras from non-orientable surfaces". Journal of Algebraic Combinatorics 42, n.º 2 (7 de março de 2015): 429–72. http://dx.doi.org/10.1007/s10801-015-0586-1.

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31

Tesler, Glenn. "Matchings in Graphs on Non-orientable Surfaces". Journal of Combinatorial Theory, Series B 78, n.º 2 (março de 2000): 198–231. http://dx.doi.org/10.1006/jctb.1999.1941.

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32

Mitchell, Jane M. O. "The Genus of the Coxeter Graph". Canadian Mathematical Bulletin 38, n.º 4 (1 de dezembro de 1995): 462–64. http://dx.doi.org/10.4153/cmb-1995-067-7.

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AbstractIn [1], Biggs stated that the Coxeter graph can be embedded in an orientable surface of genus 3. The purpose of this note is to point out that Biggs' embedding is in fact into a non-orientable surface. Further, it is shown that the orientable genus is 3 and the non-orientable genus is 6.

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33

Teragaito, Masakazu. "On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces". Canadian Mathematical Bulletin 49, n.º 4 (1 de dezembro de 2006): 624–27. http://dx.doi.org/10.4153/cmb-2006-057-5.

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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.

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34

Costa, Antonio F., Milagros Izquierdo e Ana Maria Porto. "On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary". International Journal of Mathematics 28, n.º 05 (17 de abril de 2017): 1750038. http://dx.doi.org/10.1142/s0129167x17500380.

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In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus [Formula: see text] with one boundary component is connected and in the case of non-orientable Klein surfaces it has [Formula: see text] components, if [Formula: see text] is odd, and [Formula: see text] components for even [Formula: see text]. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.

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35

Sidorov, Sergey V., e Ekaterina E. Chilina. "On non-hyperbolic algebraic automorphisms of a two-dimensional torus". Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 23, n.º 3 (30 de setembro de 2021): 295–307. http://dx.doi.org/10.15507/2079-6900.23.202103.295-307.

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Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

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36

Johnson, F. E. A. "A class of non-Kählerian manifolds". Mathematical Proceedings of the Cambridge Philosophical Society 100, n.º 3 (novembro de 1986): 519–21. http://dx.doi.org/10.1017/s030500410006624x.

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Let S+ (resp. S−) denote the class of fundamental groups of closed orientable (resp. non-orientable) 2-manifolds of genus ≥ 2, and let surface = S+ ∪ S−. In the list of problems raised at the 1977 Durham Conference on Homological Group Theory occurs the following([7], p. 391, (G. 3)).

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37

COSTA, ANTONIO F., WENDY HALL e DAVID SINGERMAN. "DOUBLES OF KLEIN SURFACES". Glasgow Mathematical Journal 54, n.º 3 (30 de março de 2012): 507–15. http://dx.doi.org/10.1017/s0017089512000109.

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Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

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38

Antolín, Yago, Warren Dicks e Peter A. Linnell. "Non-orientable surface-plus-one-relation groups". Journal of Algebra 326, n.º 1 (janeiro de 2011): 4–33. http://dx.doi.org/10.1016/j.jalgebra.2009.03.001.

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39

WOOD, JOHN C. "HARMONIC MORPHISMS AND HERMITIAN STRUCTURES ON EINSTEIN 4-MANIFOLDS". International Journal of Mathematics 03, n.º 03 (junho de 1992): 415–39. http://dx.doi.org/10.1142/s0129167x92000187.

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We show that a submersive harmonic morphism from an orientable Einstein 4-manifold M4 to a Riemann surface, or a conformal foliation of M4 by minimal surfaces, determines an (integrable) Hermitian structure with respect to which it is holomorphic. Conversely, any nowhere-Kähler Hermitian structure of an orientable anti-self-dual Einstein 4-manifold arises locally in this way. In the case M4=ℝ4 we show that a Hermitian structure, viewed as a map into S2, is a harmonic morphism; in this case and for S4, [Formula: see text] we determine all (submersive) harmonic morphisms to surfaces locally, and, assuming a non-degeneracy condition on the critical points, globally.

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40

Auckly, David, e Rustam Sadykov. "On non‐orientable surfaces embedded in 4‐manifolds". Journal of Topology 14, n.º 2 (10 de maio de 2021): 587–607. http://dx.doi.org/10.1112/topo.12188.

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41

VENDRÚSCOLO, D. "BOUNDS ON COINCIDENCE INDICES ON NON-ORIENTABLE SURFACES". Chinese Annals of Mathematics 26, n.º 02 (abril de 2005): 315–22. http://dx.doi.org/10.1142/s0252959905000257.

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42

Gadgil, Siddhartha, e Dishant Pancholi. "Homeomorphisms and the homology of non-orientable surfaces". Proceedings Mathematical Sciences 115, n.º 3 (agosto de 2005): 251–57. http://dx.doi.org/10.1007/bf02829656.

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43

Norbury, Paul. "Lengths of geodesics on non-orientable hyperbolic surfaces". Geometriae Dedicata 134, n.º 1 (19 de abril de 2008): 153–76. http://dx.doi.org/10.1007/s10711-008-9251-3.

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44

Bujalance, E., J. M. Gamboa e C. Maclachlan. "Minimum topological genus of compact bordered Klein surfaces admitting a prime-power automorphism". Glasgow Mathematical Journal 37, n.º 2 (maio de 1995): 221–32. http://dx.doi.org/10.1017/s0017089500031128.

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In the nineteenth century, Hurwitz [8] and Wiman [14] obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, [11], [12]. These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey [7] and Maclachlan [10], respectively, in the case of Riemann surfaces and by Bujalance [2], Hall [6] and Gromadzki [5] in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf [1])—was minimized by Bujalance- Etayo-Gamboa-Martens [3] in the case where G is cyclic and by McCullough [13] in the abelian case.

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45

Szepietowski, Błażej. "Mapping class group of a non-orientable surface and moduli space of Klein surfaces". Comptes Rendus Mathematique 335, n.º 12 (dezembro de 2002): 1053–56. http://dx.doi.org/10.1016/s1631-073x(02)02617-1.

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46

GRIGGS, TERRY S., CONSTANTINOS PSOMAS e JOZEF ŠIRÁŇ. "MAXIMUM GENUS EMBEDDINGS OF LATIN SQUARES". Glasgow Mathematical Journal 60, n.º 2 (30 de outubro de 2017): 495–504. http://dx.doi.org/10.1017/s0017089517000234.

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AbstractIt is proved that every non-trivial Latin square has an upper embedding in a non-orientable surface and every Latin square of odd order has an upper embedding in an orientable surface. In the latter case, detailed results about the possible automorphisms and their actions are also obtained.

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47

Fine, Benjamin, e Gerhard Rosenberger. "Surface groups within Baumslag doubles". Proceedings of the Edinburgh Mathematical Society 54, n.º 1 (19 de janeiro de 2011): 91–97. http://dx.doi.org/10.1017/s0013091509001102.

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AbstractA conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

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Satoh, Shin. "Triple point invariants of non-orientable surface-links". Topology and its Applications 121, n.º 1-2 (junho de 2002): 207–18. http://dx.doi.org/10.1016/s0166-8641(01)00118-3.

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Kuno, Erika. "Uniform hyperbolicity for curve graphs of non-orientable surfaces". Hiroshima Mathematical Journal 46, n.º 3 (novembro de 2016): 343–55. http://dx.doi.org/10.32917/hmj/1487991626.

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Yoshiji, Katsuhiro. "On the first eigenvalue of non-orientable closed surfaces". Tsukuba Journal of Mathematics 22, n.º 3 (dezembro de 1998): 741–46. http://dx.doi.org/10.21099/tkbjm/1496163676.

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