scholarly journals Twistor Bundle of a Neutral Kähler Surface

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 43
Author(s):  
Włodzimierz Jelonek
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Complex Structure ◽  
Killing Vector ◽  
Kähler Manifold ◽  
Killing Vector Field ◽  
Kahler Manifold ◽  
Kähler Surface ◽  
Kähler Surfaces ◽  
Twistor Bundle

In this paper, we characterize neutral Kähler surfaces in terms of their positive twistor bundle. We prove that an O+,+(2,2)-oriented four-dimensional neutral semi-Riemannian manifold (M,g) admits a complex structure J with ΩJ∈⋀−M, such that (M,g,J) is a neutral-Kähler manifold if and only if the twistor bundle (Z1(M),gc) admits a vertical Killing vector field.

Conformal vector fields on almost co-Kähler manifolds

2021 ◽  
Vol 71 (6) ◽  
pp. 1545-1552
Author(s):  
Uday Chand De ◽  
Young Jin Suh ◽  
Sudhakar K. Chaubey
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Killing Vector Field ◽  
Kahler Manifolds ◽  
Conformal Vector Field ◽  
Kahler Manifold ◽  
Conformal Vector ◽  
Conformal Vector Fields

Abstract In this paper, we characterize almost co-Kähler manifolds with a conformal vector field. It is proven that if an almost co-Kähler manifold has a conformal vector field that is collinear with the Reeb vector field, then the manifold is a K-almost co-Kähler manifold. It is also shown that if a (κ, μ)-almost co-Kähler manifold admits a Killing vector field V, then either the manifold is K-almost co-Kähler or the vector field V is an infinitesimal strict contact transformation, provided that the (1,1) tensor h remains invariant under the Killing vector field.

scholarly journals Holomorphic 2-Forms and Vanishing Theorems for Gromov–Witten Invariants

2009 ◽  
Vol 52 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Junho Lee
Keyword(s):  
Complex Structure ◽  
Kähler Manifold ◽  
Higher Dimensions ◽  
Almost Complex Structure ◽  
Vanishing Theorems ◽  
Kahler Manifold ◽  
Almost Complex ◽  
Kähler Surfaces ◽  
Compact Kähler Manifold

AbstractOn a compact Kähler manifold X with a holomorphic 2-form α, there is an almost complex structure associated with α. We show how this implies vanishing theorems for the Gromov–Witten invariants of X. This extends the approach used by Parker and the author for Kähler surfaces to higher dimensions.

scholarly journals On the Classification of ALE Kähler Manifolds

2020 ◽  
Author(s):  
Hans-Joachim Hein ◽  
Rareş Răsdeaconu ◽  
Ioana Şuvaina
Keyword(s):  
Complex Structure ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Quotient Singularity ◽  
Kahler Manifold ◽  
Kähler Surfaces

Abstract The underlying complex structure of an ALE Kähler manifold is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exist only finitely many diffeomorphism types of minimal ALE Kähler surfaces with a given group at infinity.

scholarly journals On Jacobi-Type Vector Fields on Riemannian Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1139 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh ◽  
Amira A. Ishan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Sufficient Conditions ◽  
Killing Vector Field ◽  
Natural Question ◽  
Necessary And Sufficient

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

scholarly journals On Affine Transformations of a Riemannian Manifold

1955 ◽  
Vol 9 ◽  
pp. 99-109 ◽  
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Affine Transformation ◽  
Integral Formula ◽  
Killing Vector ◽  
Decomposition Theorem ◽  
Complete Riemannian Manifold ◽  
Killing Vector Field ◽  
Affine Transformations ◽  
Infinitesimal Affine Transformation

In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

scholarly journals Fixed Points of Isometries

1958 ◽  
Vol 13 ◽  
pp. 63-68 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Fixed Points ◽  
Killing Vector ◽  
Connected Components ◽  
Killing Vector Field ◽  
Set Of Points ◽  
Infinitesimal Isometry

The purpose of this paper is to prove the followingTheorem. Let M be a Riemannian manifold of dimension n and let ξ be a Killing vector field (i.e., infinitesimal isometry) of M. Let F be the set of points x of M where ξ vanishes and let F = ∪ Vi, where the Vi’s are the connected components of F. Then (assuming F to be non-empty)

scholarly journals A Note on Killing Calculus on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 307
Author(s):  
Sharief Deshmukh ◽  
Amira Ishan ◽  
Suha B. Al-Shaikh ◽  
Cihan Özgür
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Laplace Operator ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Adjoint Operator ◽  
Dimensional Sphere ◽  
Killing Vector Field ◽  
Smooth Functions ◽  
The Laplace Operator

In this article, it has been observed that a unit Killing vector field ξ on an n-dimensional Riemannian manifold (M,g), influences its algebra of smooth functions C∞(M). For instance, if h is an eigenfunction of the Laplace operator Δ with eigenvalue λ, then ξ(h) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian Hh(ξ,ξ) of a smooth function h∈C∞(M) defines a self adjoint operator ⊡ξ and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold (M,g). We study several properties of functions associated to the unit Killing vector field ξ. Finally, we find characterizations of the odd dimensional sphere using properties of the operator ⊡ξ and the nontrivial solution of Fischer–Marsden differential equation, respectively.

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