Blind Peer Reviewed Refereed Open Access International Journal
8
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com HERMITIAN MANIFOLD GEOMETRY AND ITS EFFECTIVE IMPORTANCE: A
STUDY
Poonam Kumari1, Dr.Sudesh Kumar2
Department of Mathematics
1,2
OPJS University, Churu (Rajasthan) – India
Abstract
We talked about in the above article, on Hermitian manifolds, the second Ricci curvature tensors of different metric associations are firmly identified with the geometry of Hermitian manifolds. A characteristic thought is to characterize a stream by utilizing second Ricci curvature tensors of different metric associations. We depict it in the accompanying. We consider a few unique Hermitian manifolds. A fascinating class of Hermitian manifolds is the reasonable Hermitian manifolds, i.e., Hermitian manifolds with coclosed K¨ahler shapes. It is outstanding that each K¨ahler manifold is adjusted. In certain literary works, they are additionally called semi-K¨ahler manifolds. In complex dimension 1 and 2, each reasonable Hermitian manifold is K¨ahler. Nonetheless, in higher dimensions, there exist non-K¨ahler manifolds which concede adjusted Hermitian metrics. First Ricci-Chern curvature and the second Ricci-Chern curvature of a Hermitian manifold can't be looked at, we can't derive that the manifold M is K¨ahler, regardless of whether the second Chern curvature is sure all over the place. As a rule, the first Ricci-Chern curvature is d-shut yet the second Ricci-Ricci-Chern curvature isn't d-shut thus they are in the extraordinary (d, ∂, ∂)- cohomology classes. For example, the Hopf manifold S 2n+1 × S 1 with standard Hermitian metric has carefully positive second Ricci-Chern curvature and nonnegative first Ricci-Chern curvature, however it is non-K¨ahler
1. OVERVIEW
In mathematics, and more explicitly in differential geometry, a Hermitian manifold is the complex simple of a Riemannian manifold. More exactly, a Hermitian manifold is a complex manifold with an easily changing Hermitian internal item on each (holomorphic) tangent space. One can likewise characterize a Hermitian manifold as a real manifold with a Riemannian metric that jam a complex structure.
Blind Peer Reviewed Refereed Open Access International Journal
9
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com covers a wide extent of topics, for instance, Sasakian geometry, Kaehler and locally conformal Kaehler geometry, the tangential CR equations, Lorentzian geometry, holomorphic true manifolds, and Para quaternionic CR submanifolds. Arranged as a tribute to Professor Aurel Bejancu, who considered the possibility of a CR submanifold of a Hermitian complex in 1978, the book gives an extraordinary layout of a couple of focuses in the geometry of CR submanifolds. Appearing by point information on the most recent advances in the region, it addresses an accommodating resource for mathematicians and physicists alike. This workshop, upheld by AIM and the NSF, will be devoted to numerical examination and differential geometry on pseudohermitian manifolds.
A complex structure is basically a practically complex structure with an integrability condition, and this condition yields a unitary structure (U(n) structure) on the manifold. By dropping this condition, we get a nearly Hermitian manifold.
On any nearly Hermitian manifold, we can present a key 2-structure (or cosymplectic structure) that depends just on the picked metric and the practically complex structure. This structure is dependably non-degenerate. With the additional integrability condition that it is shut (i.e., it is a symplectic structure), we get a nearly Kähler structure. On the off chance that both the practically complex structure and the central structure are integrable, at that point we have a Kähler structure.
Definition
“A Hermitian metric on a complex vector bundle E over a smooth manifold M is a smoothly varying positive-definite Hermitian form on each fiber. Such a metric can be written as a smooth section”
such that
for all ζ, η in Ep and
Blind Peer Reviewed Refereed Open Access International Journal
10
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com “A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent space. Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent space”.
On a Hermitian manifold the metric can be written in local holomorphic coordinates (zα) as
where are the components of a positive-definite Hermitian matrix?
Riemannian metric and associated form
“A Hermitian metric h on an (almost) complex manifold M defines a Riemannian metric g on the underlying smooth manifold. The metric g is defined to be the real part of h”:
“The form g is a symmetric bilinear form on TMC, the complexified tangent bundle. Since g is equal to its conjugate it is the complexification of a real form on TM. The symmetry and positive definiteness of g on TM follow from the corresponding properties of h. In local holomorphic coordinates the metric g can be written”
“One can also associate to h a complex differential form ω of degree (1,1). The form ω is defined as minus the imaginary part of h”:
Once more, since ω is equivalent to its conjugate it is the complexification of a real structure on TM. The structure ω is called differently the related (1,1) structure, the central structure, or the Hermitian structure. In nearby holomorphic facilitates ω can be composed
Blind Peer Reviewed Refereed Open Access International Journal
11
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com for all complex tangent vectors u and v. The Hermitian metric h can be recouped from g and ω by means of the identity
“All three forms h, g, and ω preserve the almost complex structure J. That is”,
for all complex tangent vectors u and v.
“A Hermitian structure on an (almost) complex manifold M can therefore be specified by either”
“a Hermitian metric h as above”,
“a Riemannian metric g that preserves the almost complex structure J, or”
“a nondegenerate 2-form ω which preserves J and is positive-definite in the sense that ω(u, Ju) > 0 for all nonzero real tangent vectors u”.
Note that many authors call g itself the Hermitian metric.
Properties
Each (nearly) complex manifold concedes a Hermitian metric. This pursues legitimately from the closely resembling explanation for Riemannian metric. Given a self-assertive Riemannian metric g on a practically complex manifold M one can build another metric g′ good with the practically complex structure J in a conspicuous way:
Blind Peer Reviewed Refereed Open Access International Journal
12
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com Each nearly Hermitian manifold M has a sanctioned volume structure which is only the Riemannian volume structure dictated by g. This structure is given as far as the related (1,1)- structure ω by
where ωn is simply the wedge result of ω n times. The volume structure is hence a real (n,n)- structure on M. In nearby holomorphic coordinates the volume structure is given by
One can also consider a hermitian metric on a holomorphic vector bundle.
Kähler manifolds
The most significant class of Hermitian manifolds are Kähler manifolds. These are Hermitian manifolds for which the Hermitian structure ω is shut:
In this case the structure ω is known as a Kähler structure. A Kähler structure is a symplectic structure, thus Kähler manifolds are normally symplectic manifolds.
A nearly Hermitian manifold whose related (1,1)- structure is shut is normally called a nearly Kähler manifold. Any symplectic manifold concedes a perfect practically complex structure making it into a nearly Kähler manifold.
Integrability
A Kähler manifold is a nearly Hermitian manifold fulfilling an integrability condition. This can be expressed in a few comparable ways.
Let (M, g, ω, J) be a nearly Hermitian manifold of real dimension 2n and let ∇ be the Levi-Civita association of g. Coming up next are comparable conditions for M to be Kähler:
ω is closed and J is integrable ∇J = 0,
∇ω = 0,
the holonomy group of ∇ is contained in the unitary group U(n) associated to J.
Blind Peer Reviewed Refereed Open Access International Journal
13
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com Specifically, if M is a Hermitian manifold, the condition dω = 0 is proportionate to the obviously a lot of more grounded conditions ∇ω = ∇J = 0. The lavishness of Kähler theory is expected partially to these properties.
2. GEOMETRY OF HERMITIAN MANIFOLDS AND ITS IMPORTANCE Hermitian manifolds by using the second Ricci curvature tensor
On Hermitian manifolds, the second Ricci curvature tensors of different metric associations are firmly identified with the geometry of Hermitian manifolds. By refining the Bochner recipes for any Hermitian complex vector bundle (Riemannain real vector bundle) with a discretionary metric association over a minimal Hermitian manifold, we can derive different disappearing theorems for Hermitian manifolds and complex vector bundles continuously Ricci curvature tensors. We will likewise present a characteristic geometric stream on Hermitian manifolds by utilizing the second Ricci curvature tensor.
It is well-known([1]) that on a minimized K¨ahler manifold, on the off chance that the Ricci curvature is sure, at that point the first Betti number is zero; in the event that the Ricci curvature is negative, at that point there is no holomorphic vector field. The key element for the proofs of such results is the K¨ahler symmetry. Then again, on a Hermitian manifold, we don't have such symmetry and there are a few distinctive Ricci curvatures.
In this part, we examine the nonexistence of holomorphic and consonant sections of a theoretical vector bundle over a reduced Hermitian manifold. Let E be a holomorphic vector bundle over a conservative Hermitian manifold (M, ω). Since the holomorphic section space H0 (M, E) is independent on the associations of E, we can pick any association on it. As we referenced over, the key part, is the second Ricci curvature of the association. For instance, on the holomorphic tangent bundle T 1,0M of a Hermitian manifold M, there are three normal associations
(1) “the complexified Levi-Civita connection ∇ on T 1,0M;” (2) “the Chern connection ∇CH on T 1,0M;”
(3) “the Bismut connection ∇B on T 1,0M”
It is outstanding that if M is K¨ahler, every one of the three associations are the equivalent. Notwithstanding, all in all, the relations among them are to some degree puzzling. In this paper, we derive certain relations about their curvatures on certain Hermitian manifolds.
Theorem 1.1.
Blind Peer Reviewed Refereed Open Access International Journal
14
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com
1. “If the second Hermitian-Ricci curvature is nonpositive everywhere, then every
∂E-closed section of E is parallel, i.e. ;”
2. “If the second Hermitian-Ricci curvature is nonpositive everywhere and negative at
some point, then ”
3. “If the second Hermitian-Ricci curvature is p-nonpositive everywhere and p-negative
at some point, then for any p ≤ q ≤ rank(E).”
3. CONCLUSION
We consider a few unique Hermitian manifolds. A fascinating class of Hermitian manifolds is the reasonable Hermitian manifolds, i.e., Hermitian manifolds with coclosed K¨ahler shapes. It is outstanding that each K¨ahler manifold is adjusted. In certain literary works, they are additionally called semi-K¨ahler manifolds. In complex dimension 1 and 2, each reasonable Hermitian manifold is K¨ahler.
Presently we consider a few uncommon Hermitian manifolds. An intriguing class of Hermitian manifolds is the reasonable Hermitian manifolds, i.e., Hermitian manifolds with coclosed K¨ahler shapes. It is outstanding that each K¨ahler manifold is adjusted. In certain literary works, they are likewise called semi-K¨ahler manifolds. In complex dimension 1 and 2, each fair Hermitian manifold is K¨ahler. In any case, in higher dimensions, there exist non-K¨ahler manifolds which concede adjusted Hermitian metrics. Such examples were developed by E. Calabi([2]), see likewise [3, 4]. There are additionally some other significant classes of non-K¨ahler adjusted manifolds, for example, complex solvmanifolds, 1-dimensional groups of K¨ahler manifolds (see [3]) and conservative complex parallelizable manifolds (aside from complex torus) (see [4]). Then again, Alessandrini-Bassaneli( [5]) demonstrated that each Moishezon manifold is adjusted thus adjusted manifolds can be developed from K¨ahler manifolds by change. For more examples, see [3, 6-8].
Each decent metric ω is a Gauduchon metric. Indeed, d ∗ω = 0 is proportional to dωn−1 = 0 thus ∂∂ωn−1 = 0. By [9], each Hermitian manifold has a Gauduchon metric. Be that as it may, there are numerous manifolds which can not bolster adjusted metrics. For instance, the Hopf surface S 3 × S 1 is non-K¨ahler, so it has no reasonable metric.
Blind Peer Reviewed Refereed Open Access International Journal
15
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com REFERENCES
[1].Bochner, S. Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52, 776-797
(1946).
[2].Calabi, E.; Eckmann, B. A class of compact, complex manifolds which are not algebraic.
Ann. of Math. (2) 58, (1953) 494-500.
[3].Michelson, M.L. On the existence of special metrics in complex geometry, Acta Math.
143 (1983), 261-295.
[4].Urakawa, H. Complex Laplacians on compact complex homogeneous spaces. J. Math.
Soc. Japan 33 (1981), no. 4, 619-638
[5].Alessandrini, L.; Bassaneli, G. Metric properties of manifolds bimeromorphic to compact
Kaehler manifolds, J. Diff. Geometry 37 (1993), 95-121.
[6].Alessandrini, L.; Bassanelli, G. A class of balanced manifolds. Proc. Japan Acad. Ser. A
Math. Sci. 80 (2004), no. 1, 6-7.
[7].Ganchev, G.; Ivanov, S. Holomorphic and Killing vector fields on compact balanced
Hermitian manifolds. Internat. J. Math. 11 (2000), no. 1, 15-28.
[8].Ganchev, G.; Ivanov, S. Harmonic and holomorphic 1-forms on compact balanced
Hermitian manifolds. Differential Geom. Appl. 14 (2001), no. 1, 79-93.
[9].Gauduchon, P. La 1-forme de torsion d’une vari´et´e hermitienne compacte. Math. Ann.