相关论文: Relating the curvature tensor and the complex Jaco…

Let M be a pseudo-Riemannian manifold with a pseudo-Hermitian complex structure $J$. We give necessary and sufficient conditions that the curvature operator $R(\pi)$ is complex linear when $\pi$ is a $J$ invariant real 2 plane. Under this…

We study the spectral geometry of the conformal Jacobi operator on a 4-dimensional Riemannian manifold (M,g). We show that (M,g) is conformally Osserman if and only if (M,g) is self-dual or anti self-dual. Equivalently, this means that the…

In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_{\xi}$ and…

Let $J(\pi)$ be the higher order Jacobi operator. We study algebraic curvature tensors where $J(\pi)J(\pi^{\perp})=J(\pi^{\perp})J(\pi)$. In the Riemannian setting, we give a complete characterization of such tensors; in the…

We use the natural lifts of the fundamental tensor field g to the cotangent bundle T*M of a Riemannian manifold (M,g), in order to construct an almost Hermitian structure (G,J) of diagonal type on T*M. The obtained almost complex structure…

Let (M,g,J) be a compact Hermitian manifold with a smooth boundary. Let $\Delta_p$ and $D_p$ be the realizations of the real and complex Laplacians on p forms with either Dirichlet or Neumann boundary conditions. We generalize previous…

Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$.…

We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the…

It is well known that the Jacobi operators completely determine the curvature tensor. The question of existence of a curvature tensor for given Jacobi operators naturally arises, which is considered and solved in the previous work.…

Let $M$ be a real hypersurface in complex projective space. The almost contact metric structure on $M$ allows us to consider, for any nonnull real number $k$, the corresponding $k$-th generalized Tanaka-Webster connection on $M$ and,…

We use some natural lifts defined on the cotangent bundle T*M of a Riemannian manifold (M,g) in order to construct an almost Hermitian structure (G,J) of diagonal type. The obtained almost complex structure J on T*M is integrable if and…

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with…

In this paper, we study Jacobi operators associated to algebraic curvature maps (tensors) on lightlike submanifolds M. We investigate conditions for an induced Rie- mann curvature tensor to be an algebraic curvature tensor on M. We…

Let (M,g) be a Riemannian manifold and G a nondegenerate g-natural metric on its tangent bundle T M . In this paper we establish a relation between the Jacobi operators of (M,g) and that of (T M,G). In the case of a Riemannian surface…

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the…

Let M be a Riemannian manifold and R its curvature tensor. For a unit vector X tangent to M at a point p, the Jacobi operator is defined by R_X = R(X, .) X$. The manifold M is called pointwise Osserman if, for every point p, the spectrum of…

Let $M$ be a real hypersurface of a complex space form with almost contact metric structure $(\phi, \xi, \eta, g)$. In this paper, we study real hypersurfaces in a complex space form whose structure Jacobi operator $R_\xi=R(\cdot,\xi)\xi$…

We introduce a new potential characterization of Osserman algebraic curvature tensors. An algebraic curvature tensor is Jacobi-orthogonal if $\mathcal{J}_XY\perp\mathcal{J}_YX$ holds for all $X\perp Y$, where $\mathcal{J}$ denotes the…

We study the geometry of pseudo-Riemannian manifolds which are Jacobi--Tsankov, i.e. J(x)J(y)=J(y)J(x) for all tangent vectors x and y. We also study manifolds which are 2-step Jacobi nilpotent, i.e. J(x)J(y)=0 for all tangent vectors x and…

We characterize Riemannian manifolds of constant sectional curvature in terms of commutation properties of their Jacobi operators.