scholarly journals Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

2017 ◽  
Vol 28 (09) ◽  
pp. 1740004 ◽  
Author(s):  
Antonio Alarcón ◽  
Finnur Lárusson
Keyword(s):  
Riemann Surface ◽  
Convex Hull ◽  
Minimal Surface ◽  
Cohomology Class ◽  
Holomorphic Maps ◽  
De Rham Cohomology ◽  
Surface Theory ◽  
Open Riemann Surface ◽  
De Rham ◽  
Smooth Locus

Let [Formula: see text] be a connected open Riemann surface. Let [Formula: see text] be an Oka domain in the smooth locus of an analytic subvariety of [Formula: see text], [Formula: see text], such that the convex hull of [Formula: see text] is all of [Formula: see text]. Let [Formula: see text] be the space of nondegenerate holomorphic maps [Formula: see text]. Take a holomorphic 1-form [Formula: see text] on [Formula: see text], not identically zero, and let [Formula: see text] send a map [Formula: see text] to the cohomology class of [Formula: see text]. Our main theorem states that [Formula: see text] is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on [Formula: see text] can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

scholarly journals NONTRIVIAL CLASSES IN H*(Imb(S1,ℝn)) FROM NONTRIVALENT GRAPH COCYCLES

2004 ◽  
Vol 01 (05) ◽  
pp. 639-650 ◽  
Author(s):  
RICCARDO LONGONI
Keyword(s):  
Linear Combination ◽  
Cohomology Class ◽  
Feynman Diagrams ◽  
De Rham Cohomology ◽  
De Rham

We construct nontrivial cohomology classes of the space Imb (S1,ℝn) of imbeddings of the circle into ℝn by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we construct, for every even number n≥4, a de Rham cohomology class on Imb (S1,ℝn). We prove nontriviality of these classes by evaluation on the dual cycles.

scholarly journals A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ali H. Alkhaldi ◽  
Aliya Naaz Siddiqui ◽  
Kamran Ahmad ◽  
Akram Ali
Keyword(s):  
Cohomology Class ◽  
Warped Product ◽  
Sasakian Manifold ◽  
Sasakian Manifolds ◽  
De Rham Cohomology ◽  
De Rham

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.

scholarly journals Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve

2018 ◽  
Vol 2018 (740) ◽  
pp. 77-109 ◽  
Author(s):  
Antonio Alarcón ◽  
Franc Forstnerič
Keyword(s):  
Riemann Surface ◽  
Minimal Surface ◽  
Minimal Immersion ◽  
Open Riemann Surface ◽  
The Real ◽  
Conformal Minimal Immersion ◽  
Null Curve ◽  
Minimal Immersions ◽  
Conformal Minimal Surface

Abstract We show that for every conformal minimal immersion {u:M\to\mathbb{R}^{3}} from an open Riemann surface M to {\mathbb{R}^{3}} there exists a smooth isotopy {u_{t}:M\to\mathbb{R}^{3}} ( {t\in[0,1]} ) of conformal minimal immersions, with {u_{0}=u} , such that {u_{1}} is the real part of a holomorphic null curve {M\to\mathbb{C}^{3}} (i.e. {u_{1}} has vanishing flux). If furthermore u is nonflat, then {u_{1}} can be chosen to have any prescribed flux and to be complete.

COHOMOLOGY OF LAGRANGE COMPLEXES INVARIANT UNDER PSEUDOGROUPS OF LOCAL TRANSFORMATIONS

2007 ◽  
Vol 04 (04) ◽  
pp. 669-705 ◽  
Author(s):  
ANDREA SPIRO
Keyword(s):  
Inverse Problem ◽  
Calculus Of Variations ◽  
Configuration Space ◽  
Cohomology Class ◽  
Cohomology Groups ◽  
Lagrange Equations ◽  
De Rham Cohomology ◽  
Base Manifold ◽  
De Rham

The inverse problem of the Calculus of Variations for Lagrangians and Euler–Lagrange equations invariant under a pseudogroup [Formula: see text] of local transformations of the base manifold is considered. Exploiting some ideas of Krupka, a theorem is proved showing that, if the configuration space consists of sections of tensor bundles or of local maps of a manifold into another, then such inverse problem is solvable whenever a certain cohomology class of [Formula: see text]-invariant forms on the configuration space is vanishing. In addition, for a few pseudogroups, the cohomology groups considered in the main result are explicitly determined in terms of the de Rham cohomology of the configuration space.

scholarly journals Study on De Rham Cohomology Algebra of Manifolds

2016 ◽  
Vol 64 (2) ◽  
pp. 109-113
Author(s):  
Saraban Tahora ◽  
Khondokar M Ahmed
Keyword(s):  
Cohomology Class ◽  
Homology Class ◽  
Cohomology Algebra ◽  
Graded Algebra ◽  
Exterior Derivative ◽  
De Rham Cohomology ◽  
Singular Homology ◽  
Orientable Manifold ◽  
De Rham ◽  
Smooth Map

In the present paper some aspects of exterior derivative, graded algebra, cohomology algebra, de Rham cohomology algebra, singular homology, cohomology class are studied. Graded subspace, smooth map, a singular P- - simplex in a manifold M, oriented n- manifold M, the space of P- cycles and P- boundaries, Pth singular homology and homology class are treated in our paper. A theorem 3.03 is established which is related to orientable manifold. Dhaka Univ. J. Sci. 64(2): 109-113, 2016 (July)

THE DE RHAM COHOMOLOGY OF DIFFERENTIAL SPACES

1989 ◽  
Vol 22 (1) ◽  
pp. 249-272 ◽  
Author(s):  
Wiesław Sasin
Keyword(s):  
De Rham Cohomology ◽  
De Rham ◽  
Differential Spaces

scholarly journals STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

2021 ◽  
pp. 1-48
Author(s):  
Federico Scavia
Keyword(s):  
Finite Groups ◽  
Reductive Groups ◽  
Orthogonal Groups ◽  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
Steenrod Operations ◽  
De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

scholarly journals Cycles in the de Rham cohomology of abelian varieties over number fields

2018 ◽  
Vol 154 (4) ◽  
pp. 850-882
Author(s):  
Yunqing Tang
Keyword(s):  
Endomorphism Ring ◽  
Abelian Varieties ◽  
Number Fields ◽  
De Rham Cohomology ◽  
Tate Conjecture ◽  
De Rham ◽  
Prime Dimension

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of$\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

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