Differential Geometry and its Applications最新文献

In this short note we provide some results for Bach solitons under different assumptions. In fact, under either non-negative or non-positive Ricci curvature condition we are able to show that a Bach soliton must be Bach-flat, since it satisfies a finite weighted Dirichlet integral condition or a parabolicity condition jointly with some regularity conditions $L^{\infty }$ or $L^p$ on gradient of the potential function.

We examine properties of equidistant sets determined by nonempty disjoint compact subsets of a compact 2-dimensional Alexandrov space (of curvature bounded below). The work here generalizes many of the known results for equidistant sets determined by two distinct points on a compact Riemannian 2-manifold. Notably, we find that the equidistant set is always a finite simplicial 1-complex. These results are applied to answer an open question concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of $M^n$ modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces.

We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov–Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fréchet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

We show that given a constant rank linear map, $K:TM\to TM$, there exists a Lie algebroid with K as its anchor map, if and only if the image distribution, ImK, is involutive. As a byproduct, a new example of Lie algebroid bracket associated with a regular foliation is obtained through the projector onto the involutive distribution. The Lie algebroid bracket is not defined on the involutive distribution but on the whole space of vector fields of the manifold.

It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a differential equation, which satisfies sufficiently many symmetries and corresponding conservation laws, leads to a variational functional, whose Euler-Lagrange equation is the given differential equation. Sufficiently many symmetries means that the set of symmetry vector fields satisfy span$\{V_p:V\in \text{Sym}\}=T_{p}E$, that is, they span the tangent space $T_{p}E$ at each point p of a fiber bundle E, which describes the dependent- and independent coordinates. Higher order coordinates are described by the jet bundle $J^{k}E$ and we require no span-assumptions on $T{J}^{k}E$. Our main theorem states that Noether's theorem can be reversed in this sense for second order differential equations, or more precisely, for so-called second order source forms on $J^{2}E$, which are required to write the differential equation as a weak formulation (every Euler-Lagrange equation is derived from a first variation, that is, from a weak formulation). Counter examples show that our theorem is sharp.

Wang and Zhang in (2022) [20] and (2023) [21] characterized type (A) real hypersurfaces of the nonflat complex space forms as having transversal Killing structure Lie operator $L_{\xi }$ or contact Lie operator $L_{\xi }\varphi$. In this note, we extend the above results by showing that the class of real hypersurfaces of type (A), (B) and the ruled real hypersurfaces in the nonflat complex space forms are locally characterized by having weakly transversal Killing operator $L_{\xi }$ or $L_{\xi }\varphi$.

In this paper, we prove that the circle centered at the origin in $B^2\left({\delta }_{\xi }\right)$ is a proper maximum of the isoperimetric problem in a 2-dimensional Randers space endowed with 3-parameter family of non-locally projectively flat Finsler metrics of non-constant isotropic S-curvature.

In this paper we study some special ruled surfaces in a 3-dimensional Riemannian manifold $\overline{M}$. Given an immersed surface M into $\overline{M}$, we consider the ruled surfaces that are normal or tangent to M and give some geometric relations between them, generalizing some recent results obtained in [3], [5]. We also give some general properties on normal and tangent submanifolds of arbitrary dimension.

We consider left-invariant $G_2$-structures on 7-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket A of the corresponding Lie algebra. In those terms, we establish the algebraic condition on A for each of the possible 16-torsion classes of a $G_2$-structure. In particular, we show that four of those torsion classes are not admissible, since ${\tau }_3=0$ implies ${\tau }_0=0$. Finally, we use the above results to provide the algebraic criteria on A, satisfying the harmonic condition $\mathrm{div}T=0$, and this allows to identify which torsion classes are harmonic.

A nonlinear splitting on a fiber bundle is a generalization of an Ehresmann connection. An example is given by the homogeneous nonlinear splitting of a Finsler function on the total manifold of a fiber bundle. We show how homogeneous nonlinear splittings and nonlinear lifts can be used to construct submersions between Euclidean, Minkowski and Finsler spaces. As an application we consider a semisimple Lie algebra and use our methods to give new examples of Finsler functions on a reductive homogeneous space.