Differential Geometry and its Applications最新文献

Let M be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let $E$ be a Hermitian vector bundle over M equipped with a metric covariant derivative ∇. We consider the operator $H_{X,V}={\nabla }^{†}\nabla +{\nabla }_X+V$, where ${\nabla }^{†}$ is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of $E$, X is a smooth (real) vector field on M, and V is a fiberwise self-adjoint, smooth section of the endomorphism bundle $\mathrm{End}E$. We give a sufficient condition for the triviality of the $L^2$-kernel of $H_{X,V}$. As a corollary, putting $X\equiv 0$ and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the $L^2$-kernel of $D^2$, where D is the Dirac operator corresponding to ∇. In particular, when $E={\Lambda }_C^k{T}^{⁎}M$ and $D^2$ is the Hodge–deRham Laplacian on (complex-valued) k-forms, we recover some recent vanishing results for $L^2$-harmonic (complex-valued) k-forms.

In this paper, we investigate the Sasakian statistical structures of constant ϕ-sectional curvature based on Sasakian space forms. We obtain the classification of this kind of Sasakian statistical structures. Our classification results show that the Sasakian statistical structures of constant ϕ-sectional curvature on a Sasakian space form with dimension higher than 3 must be almost-trivial; on a 3-dimensional Sasakian space form, in addition to the almost-trivial Sasakian statistical structure, there exist other Sasakian statistical structures which satisfy the constant ϕ-sectional curvature condition. We also point out that a rigidity result for cosymplectic statistical structures of constant ϕ-sectional curvature on 3-dimensional cosymplectic space forms in [11] can be improved to the corresponding classification result.

We study the $\mathrm{SU}\left(3\right)$-structure induced on an oriented hypersurface of a 7-dimensional manifold with a nearly parallel $G_2$-structure. Such $\mathrm{SU}\left(3\right)$-structures are called nearly half-flat. We characterise the left invariant nearly half-flat structures on $S^3\times S^3$. This characterisation then helps us to systematically analyse nearly parallel $G_2$-structures on an interval times $S^3\times S^3$.

Let M be a smooth manifold and $F$ a Morse-Bott foliation with a compact critical manifold $\Sigma \subset M$. Denote by $D\left(F\right)$ the group of diffeomorphisms of M leaving invariant each leaf of $F$. Under certain assumptions on $F$ it is shown that the computation of the homotopy type of $D\left(F\right)$ reduces to three rather independent groups: the group of diffeomorphisms of Σ, the group of vector bundle automorphisms of some regular neighborhood of Σ, and the subgroup of $D\left(F\right)$ consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups $D\left(F\right)$ for such foliations are also presented.

The purpose of this paper is to clarify and extend the result of K. Okumura in [7] concerning hypersurfaces in the non-flat complex space forms $C{P}^n$ and $C{H}^n$ whose *-Ricci tensor is $D$-recurrent.

In this article, we study the Zermelo navigation problem with and without obstacles from a theoretical point of view and look towards some computational aspects. More intuitively, this navigation model is in fact an optimal control problem with continuous inequality constraints. We first aim to study the structure of these optimal trajectories using the geometric aspects of the problem. More precisely, we find the time-optimal trajectories and characterize them as geodesics of Randers metrics away from the danger zone and geodesics of (not necessarily Randers) Finsler metrics where they touch the boundary of the danger zone. We demonstrate some of the important behavior of these trajectories by examples. In particular, we will calculate these trajectories precisely for the critical case of an infinitesimal homothety which, in the language of optimal control problems, will be referred to in this paper as a weak linear vortex.

Regarding the computational aspects of the resulting optimal control problem with constraints and inspired by the geometry behind this problem, we propose a modification of the optimization scheme previously considered in [Li-Xu-Teo-Chu, Time-optimal Zermelo's navigation problem with moving and fixed obstacles, 2013] by adding a piecewise constant rotation. This modification will entail adding another piecewise constant control to the problem which in turn proves to make the resulting approximated time-optimal paths more precise and efficient as we argue by the example of navigation through a linear vortex.

In this paper, we mainly prove that on Kenmotsu and Sasakian statistical manifolds, the Riemannian curvature tensor and the statistical curvature tensor fields are equal, only if their covariant derivatives are equal.