相关论文: Harmonicity of unit vector fields with respect to …

Aguilar introduced isotropic almost complex structures $J_{\delta , \sigma}$ on the tangent bundle of a Riemannian manifold $(M, g)$. In this paper, some results will be obtained on the integrability of these structures. These structures…

Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, the only vector fields which define harmonic maps from $(M,g)$ to $(TM,g^s)$, are the parallel ones. The Sasaki…

This article deals with the interpolating sesqui-harmonicity of a vector field $X$ viewed as a map from a Riemannian manifold $(M,g)$ to its tangent bundle $TM$ endowed with the Sasaki metric $g_{S}$. We show characterization theorem for…

Let $(M,g)$ be a compact Riemannian manifold. Equipping its tangent bundle $TM$ (resp. unit tangent bundle $T_1M$) by a pseudo-Riemannian $g$-natural metric $G$ (resp. $\tilde{G}$), we study the biharmonicty of vector fields (resp. unit…

Let $(M,g)$ be a Riemannian manifold, $L(M)$ its frame bundle. We construct new examples of Riemannian metrics on $L(M)$, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi--Civita connection and…

As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian…

Let $(M,g)$ be an $n-$dimensional Riemannian manifold and $T_{1}^{1}(M)$ be its $(1,1)-$tensor bundle equipped with the rescaled Sasaki type metric $% ^{S}g_{f}$ which rescale the horizontal part by a nonzero differentiable function $f$. In…

Given a closed Riemannian manifold $(M^m,g)$ and a vector field $v$ on $M$, we form the Sasaki metric $g_S$ on $TM$, and restrict it to the image of the cross section map of $M$ into $TM$ defined by $v$, whose pull back to $M$ defines a new…

Given a flat vector bundle over a compact Riemannian manifold, Corlette and Donaldson proved that it admits harmonic metrics if and only if it is semi-simple. In this paper, we extend this equivalence to arbitrary vector bundles without any…

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively…

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected…

The bienergy of a vector field on a Riemannian manifold (M,g) is defined to be the bienergy of the corresponding map (M,g) ---> (TM,g_S), where the tangent bundle TM is equipped with the Sasaki metric g_S. The constrained variational…

This work investigates biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-K\"ahler--Norden manifold (M, varphi, g) endowed with the varphi-Sasaki metric. We derive the first variation of the bienergy…

For any flag manifold G/T we obtain an explicit expression of its Levi-Civita connection with respect to any invariant Riemannian metric.

We prove the existence and uniqueness of Levi-Civita connections for strongly sigma-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well…

This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for…

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, <,>) equipped with the Sasaki metric and we discuss the characterising condition for critical points. Likewise, we provide a useful…

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar g$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and…

Let Riemannian metrics $g$ and $\bar g$ on a connected manifold $M^n$ have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the…

Let $(\Sigma,g)$ be a closed Riemannian surface, $\textbf{G}=\{\sigma_1,\cdots,\sigma_N\}$ be an isometric group acting on it. Denote a positive integer $\ell=\inf_{x\in\Sigma}I(x)$, where $I(x)$ is the number of all distinct points of the…