scholarly journals Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1890
Author(s):  
Lucian-Miti Ionescu ◽  
Cristina-Liliana Pripoae ◽  
Gabriel-Teodor Pripoae
Keyword(s):  
Differential Geometry ◽  
Vector Field ◽  
Vector Fields ◽  
Torsion Tensor ◽  
Holomorphic Functions ◽  
Tensor Fields ◽  
Pedagogical Tool ◽  
Complex Integral ◽  
Curvature And Torsion

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

scholarly journals Clifford algebra-based structure filtering analysis for geophysical vector fields

2013 ◽  
Vol 20 (4) ◽  
pp. 563-570 ◽  
Author(s):  
Z. Yu ◽  
W. Luo ◽  
L. Yi ◽  
Y. Hu ◽  
L. Yuan
Keyword(s):  
Vector Field ◽  
Clifford Algebra ◽  
El Niño ◽  
Vector Fields ◽  
El Nino ◽  
Field Analysis ◽  
Directional Information ◽  
Filtering Method ◽  
Additional Information

Abstract. A new Clifford algebra-based vector field filtering method, which combines amplitude similarity and direction difference synchronously, is proposed. Firstly, a modified correlation product is defined by combining the amplitude similarity and direction difference. Then, a structure filtering algorithm is constructed based on the modified correlation product. With custom template and thresholds applied to the modulus and directional fields independently, our approach can reveal not only the modulus similarities but also the classification of the angular distribution. Experiments on exploring the tempo-spatial evolution of the 2002–2003 El Niño from the global wind data field are used to test the algorithm. The results suggest that both the modulus similarity and directional information given by our approach can reveal the different stages and dominate factors of the process of the El Niño evolution. Additional information such as the directional stability of the El Niño can also be extracted. All the above suggest our method can provide a new powerful and applicable tool for geophysical vector field analysis.

scholarly journals Finitely curved orbits of complex polynomial vector fields

2007 ◽  
Vol 79 (1) ◽  
pp. 13-16
Author(s):  
Albetã C. Mafra
Keyword(s):  
Vector Field ◽  
Gaussian Curvature ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Complex Polynomial ◽  
Holomorphic Foliations ◽  
Holomorphic Curve ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

This note is about the geometry of holomorphic foliations. Let X be a polynomial vector field with isolated singularities on C². We announce some results regarding two problems: 1. Given a finitely curved orbit L of X, under which conditions is L algebraic? 2. If X has some non-algebraic finitely curved orbit L what is the classification of X? Problem 1 is related to the following question: Let C <FONT FACE=Symbol>Ì</FONT> C² be a holomorphic curve which has finite total Gaussian curvature. IsC contained in an algebraic curve?

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals The Frenet Vector Fields and the Curvatures of the Natural Lift Curve

2012 ◽  
Vol 2 ◽  
pp. 38-43 ◽  
Author(s):  
Evren Ergün ◽  
Mustafa Bilici ◽  
Mustafa Çalişkan
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Similar Calculation ◽  
Darboux Vector ◽  
Lift Curve ◽  
Curvature And Torsion ◽  
The Given ◽  
Made In

In this paper, Frenet vector fields, curvature and torsion of the natural lift curve of a given curve is calculated by using the angle between Darboux vector field and the binormal vector field of the given curve in 3/1 . Also, a similar calculation is made in 3/1 considering timelike or spacelike Darboux vector field.

Differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Vector Fields ◽  
Differential Forms ◽  
Curvilinear Coordinates ◽  
Moving Frames ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Vector Calculus ◽  
Vector Version ◽  
Definition Of

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝn

2020 ◽  
Vol 28 (2) ◽  
pp. 173-184 ◽  
Author(s):  
Rohit Kumar Mishra
Keyword(s):  
Vector Field ◽  
Integral Geometry ◽  
Tensor Field ◽  
Vector Fields ◽  
First Integral ◽  
Tensor Fields ◽  
Line Complex ◽  
Full Reconstruction ◽  
Ill Posed ◽  
Fixed Curve

AbstractWe show that a vector field in {\mathbb{R}^{n}} can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve {\gamma\subset\mathbb{R}^{n}}. The question of reconstruction of a symmetric m-tensor field from the knowledge of the first {m+1} integral moments was posed by Sharafutdinov [Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser. 1, De Gruyter, Berlin, 1994, p. 78]. In this work, we provide an answer to Sharafutdinov’s question for the case of vector fields from restricted data comprising of the first two integral moment transforms.

scholarly journals CONFORMAL FIELDS: A CLASS OF REPRESENTATIONS OF Vect(N)

1992 ◽  
Vol 07 (26) ◽  
pp. 6493-6508 ◽  
Author(s):  
T.A. LARSSON
Keyword(s):  
Differential Geometry ◽  
Representation Theory ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Tensor Fields ◽  
Normal Ordering ◽  
New Class ◽  
Finite Dimensional ◽  
Conformal Fields

Vect (N), the algebra of vector fields in N dimensions, is studied. Some aspects of local differential geometry are formulated as Vect(N) representation theory. There is a new class of modules, conformal fields, whose restrictions to the subalgebra sl(N+1)⊂ Vect (N) are finite-dimensional sl (N+1) representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match.

scholarly journals About Formal Normal Form of the Semi-Hyperbolic Maps Germs on the Plane

2021 ◽  
Vol 38 ◽  
pp. 54-64
Author(s):  
P. A. Shaikhullina ◽  
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
The Other ◽  
Time Shift ◽  
Saddle Node ◽  
Dimension 2 ◽  
The One

There are consider the problem of constructing an analytical classification holomorphic resonance maps germs of Siegel-type in dimension 2. Namely, semi-hyperbolic maps of general form: such maps have one parabolic multiplier (equal to one), and the other hyperbolic (not equal in modulus to zero or one). In this paper, the first stage of constructing an analytical classification by the method of functional invariants is carried out: a theorem on the reducibility of a germ to its formal normal form by "semiformal" changes of coordinates is proved. The one-time shift along the saddle node vector field (the formal normal form in the problem of the analytical classification of saddle-node vector fields on a plane) is chosen as the formal normal form.

scholarly journals The Geometry of Tangent Bundles: Canonical Vector Fields

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Tongzhu Li ◽  
Demeter Krupka
Keyword(s):  
Vector Field ◽  
Linear Combination ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Complete Classification ◽  
Constant Coefficients ◽  
Vertical Lift ◽  
Tangent Bundles ◽  
Canonical Vector

A canonical vector field on the tangent bundle is a vector field defined by an invariant coordinate construction. In this paper, a complete classification of canonical vector fields on tangent bundles, depending on vector fields defined on their bases, is obtained. It is shown that every canonical vector field is a linear combination with constant coefficients of three vector fields: the variational vector field (canonical lift), the Liouville vector field, and the vertical lift of a vector field on the base of the tangent bundle.

scholarly journals On Some Characterizations of Vector Fields on Manifolds

2012 ◽  
Vol 60 (2) ◽  
pp. 259-263 ◽  
Author(s):  
Khondokar M. Ahmed ◽  
Md. Raknuzzaman ◽  
Md. Showkat Ali
Keyword(s):  
Differential Geometry ◽  
Vector Field ◽  
Vector Fields ◽  
Definition Of ◽  
Tangent Bundles

The basic geometry of vector fields and definition of the notions of tangent bundles are developed in an essential different way than in usual differential geometry. ø-related vector fields are studied and some related properties are developed in our paper. Finally, a theorem 5.04 on our natural injection j of submanifolds which is j-related to vector field X is treated.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11530 Dhaka Univ. J. Sci. 60(2): 259-263, 2012 (July)

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