Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe ciconia metric on the tangent bundle of an almost-Hermitian manifold
168
0
0.0
(
0
)
Authors
Rui Albuquerque
Ask ChatGPT about the research
No Arabic abstract
We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kahlerian Ricci-flat manifolds in four real dimensions.
rate research
Read More
We find a family of Kahler metrics invariantly defined on the radius $r_0>0$ tangent disk bundle ${{cal T}_{M,r_0}}$ of any given real space-form $M$ or any of its quotients by discrete groups of isometries. Such metrics are complete in the non-negative curvature case and non-complete in the negative curvature case. If $dim M=2$ and $M$ has constant sectional curvature $K eq0$, then the Kahler manifolds ${{cal T}_{M,r_0}}$ have holonomy $mathrm{SU}(2)$; hence they are Ricci-flat. For $M=S^2$, just this dimension, the metric coincides with the Stenzel metric on the tangent manifold ${{cal T}_{S^2}}$, giving us a new most natural description of this well-know metric.
We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $omega$ from the Heisenberg manifold $(H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F^H=L(E_1^H,E_2^H)$ and $F^V=L(E_1^V,E_2^V)$ are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold $(H_3,g)$, are geodesics in the tangent bundle endowed with the Sasaki metric $(TH_3,g^s)$, if and only if the curves considered on the base manifold are geodesics. Then, we get two particular examples of geodesics from $(TH_3,g^s)$, which are not horizontal or natural lifts of geodesics from the base manifold $(H_3,g)$.
We establish a new criterion for a compatible almost complex structure on a symplectic four-manifold to be integrable and hence Kahler. Our main theorem shows that the existence of three linearly independent closed J-anti-invariant two-forms implies the integrability of the almost complex structure. This proves the conjecture of Draghici-Li-Zhang in the almost-Kahler case
Let $Kbackslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Gamma ,subset, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kahler manifold and $rho, :, pi_1(X, x_0),longrightarrow, Gamma$ a homomorphism such that the adjoint action of $rho(pi_1(X, x_0))$ on $text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $rho$ a harmonic map $X, longrightarrow, Kbackslash G/Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.
We continue the study of the anti-Hermitian structures of general natural lift type on the tangent bundles. We get the conditions under which these structures are in the eight classes obtained by Ganchev and Borisov. We complete the characterization of the general natural anti-Kahlerian structures on the tangent bundles with necessary and sufficient conditions, then we present some results concerning the remaining classes.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
