Example of a six-dimensional LCK solvmanifold

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

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On locally conformal Kähler space forms

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171285000060 ◽

1985 ◽

Vol 8 (1) ◽

pp. 69-74 ◽

Cited By ~ 2

Author(s):

Koji Matsumoto

Keyword(s):

Space Form ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Covariant Differentiation ◽

Space Forms ◽

Riemannian Curvature Tensor ◽

Locally Conformal Kähler ◽

Necessary And Sufficient ◽

Locally Conformal ◽

Lee Form

Anm-dimensional locally conformal Kähler manifold (l.c.K-manifold) is characterized as a Hermitian manifold admitting a global closedl-formαλ(called the Lee form) whose structure(Fμλ,gμλ)satisfies∇νFμλ=−βμgνλ+βλgνμ−αμFνλ+αλFνμ, where∇λdenotes the covariant differentiation with respect to the Hermitian metricgμλ,βλ=−Fλϵαϵ,Fμλ=Fμϵgϵλand the indicesν,μ,…,λrun over the range1,2,…,m.For l. c. K-manifolds, I. Vaisman [4] gave a typical example and T. Kashiwada ([1], [2],[3]) gave a lot of interesting properties about such manifolds.In this paper, we shall study certain properties of l. c. K-space forms. In§2, we shall mainly get the necessary and sufficient condition that an l. c. K-space form is an Einstein one and the Riemannian curvature tensor with respect togμλwill be expressed without the tensor fieldPμλ. In§3, we shall get the necessary and sufficient condition that the length of the Lee form is constant and the sufficient condition that a compact l. c. K-space form becomes a complex space form. In the last§4, we shall prove that there does not exist a non-trivial recurrent l. c. K-space form.

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Infinitesimal Transformations of Locally Conformal Kähler Manifolds

Mathematics ◽

10.3390/math7080658 ◽

2019 ◽

Vol 7 (8) ◽

pp. 658

Author(s):

Yevhen Cherevko ◽

Volodymyr Berezovski ◽

Irena Hinterleitner ◽

Dana Smetanová

Keyword(s):

Sufficient Conditions ◽

Lie Derivative ◽

Conformal Transformations ◽

Projective Transformations ◽

Integrability Conditions ◽

Locally Conformal Kähler ◽

Necessary And Sufficient ◽

Locally Conformal ◽

Lee Form ◽

Infinitesimal Transformations

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric.

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Locally Conformal Kähler and Hermitian Yang-Mills Metrics

Chinese Annals of Mathematics Series B ◽

10.1007/s11401-021-0274-5 ◽

2021 ◽

Vol 42 (4) ◽

pp. 511-518

Author(s):

Jieming Yang

Keyword(s):

Yang Mills ◽

Locally Conformal Kähler ◽

Locally Conformal

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Topological equivalence and rigidity of flows on certain solvmanifolds

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385799130116 ◽

1999 ◽

Vol 19 (3) ◽

pp. 559-569

Author(s):

D. BENARDETE ◽

S. G. DANI

Keyword(s):

Lie Group ◽

Vector Spaces ◽

Complex Numbers ◽

Real Vector ◽

Orbit Equivalent ◽

Finite Dimensional ◽

Generic Class ◽

Induced Flow ◽

Simply Connected ◽

Solvable Lie Group

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

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Singularity of the varieties of representations of lattices in solvable Lie groups

Journal of Topology and Analysis ◽

10.1142/s1793525316500114 ◽

2016 ◽

Vol 08 (02) ◽

pp. 273-285 ◽

Cited By ~ 1

Author(s):

Hisashi Kasuya

Keyword(s):

Lie Group ◽

Vector Bundles ◽

Similar Technique ◽

Trivial Representation ◽

Nilpotent Lie Algebra ◽

Solvable Lie Groups ◽

Analytic Germ ◽

Space Construction ◽

Simply Connected ◽

Solvable Lie Group

For a lattice [Formula: see text] of a simply connected solvable Lie group [Formula: see text], we describe the analytic germ in the variety of representations of [Formula: see text] at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by [Formula: see text]. By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of [Formula: see text] at the trivial representation by using the Kuranishi space construction. By a similar technique, we also study deformations of holomorphic structures of trivial vector bundles over complex parallelizable solvmanifolds.

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The Ricci pinching functional on solvmanifolds

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haz020 ◽

2019 ◽

Author(s):

Jorge Lauret ◽

Cynthia E Will

Keyword(s):

Lie Groups ◽

Lie Group ◽

Invariant Metrics ◽

Solvable Lie Groups ◽

Left Invariant Metrics ◽

Solvable Lie Group

Abstract We study the natural functional $F=\frac {\operatorname {scal}^2}{|\operatorname {Ric}|^2}$ on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension $n$. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of $F$ restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group, and beyond the unimodular case, we obtain the same result for almost-abelian Lie groups. Many other aspects of the behavior of $F$ are clarified.

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A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500570 ◽

2012 ◽

Vol 09 (07) ◽

pp. 1250057 ◽

Cited By ~ 2

Author(s):

DOBRINKA GRIBACHEVA

Keyword(s):

Curvature Tensor ◽

Sufficient Conditions ◽

Flat Connection ◽

Riemannian Product ◽

Natural Connection ◽

Hermitian Geometry ◽

Locally Conformal Kähler ◽

Basic Class ◽

Necessary And Sufficient ◽

Locally Conformal

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

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