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We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=frac{1}{2}pounds _xi varphi$ and $ell := R(cdot,xi)xi$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $xi$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(mathcal{H}(M), J, theta)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,varphi,xi,eta,g)$ is Sasakian if and only if the corresponding nondegenerate almost $CR$ structure $(mathcal{H}(M), J)$ is integrable and $J$ is parallel along $xi$ with respect to the Bott partial connection.
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We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the Riemann curvature tensor and find conditions relating the harmonicity of the almost contact and almost complex structures of the total and base spaces of the Morimoto fibration.
In $N(k)$-contact metric manifolds and/or $(k,mu)$-manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with $V$ pointwise collinear with the structure vector field $xi $ are studied.
We generalize a Bernstein-type result due to Albujer and Alias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Sigma_1times mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: Sigma_1to Sigma_2$ between two Riemannian manifolds $(Sigma_1^m, g_1)$ and $(Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $Sigma_1times Sigma_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $mathrm{Ricci}_1 geq 0$ and $K_1geq K_2$, we prove that, if the second fundamental form of $M$ satisfies an integrability condition, then $M$ is totally geodesic, and it is a slice if $mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $pin Sigma_1$, $K_1(p)> 0$, and if $Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $Sigma_2$.
In this paper, we use the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtian the $C^2$-estimate of canonical second fundamental form, Liouville type theorems of pseudoholomorphic maps, pseudoholomorphicity of pluriharmonic maps, and Simons integral inequality of pseudoholomorphic isometric immersion.
In this paper, we discuss the heat flow of a pseudo-harmonic map from a closed pseudo-Hermitian manifold to a Riemannian manifold with non-positive sectional curvature, and prove the existence of the pseudo-harmonic map which is a generalization of Eells-Sampsons existence theorem. We also discuss the uniqueness of the pseudo-harmonic representative of its homotopy class which is a generalization of Hartman theorem, provided that the target manifold has negative sectional curvature.
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