Differential Geometry and its Applications最新文献

In this paper, we find a new tensor which is responsible for Finsler metrics with reversible geodesics. Using this tensor, we can prove that Finsler metrics are Douglas metrics if and only if they have reversible geodesics and Douglas curvature. Further, we focus on Finsler metrics with reversible Douglas curvature.

We study the Gromov–Hausdorff convergence of metric pairs and metric tuples and prove the equivalence of different natural definitions of this concept. We also prove embedding, completeness and compactness theorems in this setting. Finally, we get a relative version of Fukaya's theorem about quotient spaces under Gromov–Hausdorff equivariant convergence and a version of Grove–Petersen–Wu's finiteness theorem for stratified spaces.

We study geometric structures arising from Hermitian forms on linear spaces over real algebras beyond the division ones. Our focus is on the dual numbers, the split-complex numbers, and the split-quaternions. The corresponding geometric structures are employed to describe the spaces of oriented geodesics in the hyperbolic plane, the Euclidean plane, and the round 2-sphere. We also introduce a simple and natural geometric transition between these spaces. Finally, we present a projective model for the hyperbolic bidisc, that is, the Riemannian product of two hyperbolic discs.

In this paper, we achieve a Reilly type integral formula associated with the ϕ-Laplacian. As its applications, we obtain Heintze-Karcher and Minkowski type inequalities. Furthermore, almost Schur lemmas are also given. They recover the partial results of Li and Xia in [17]. On the other hand, we also study eigenvalue problem for Wentzell boundary conditions and obtain eigenvalue relationships.

We study sequences of immersions respecting bounds coming from Riemannian geometry and apply the ensuing results to the study of sequences of submanifolds of symplectic and contact manifolds. This allows us to study the subtle interaction between the Hausdorff metric and the Lagrangian Hofer and spectral metrics. In the process, we get proofs of metric versions of the nearby Lagrangian conjecture and of the Viterbo conjecture on the spectral norm. We also get $C^0$-rigidity results for a vast class of important submanifolds of symplectic and contact manifolds in the presence of Riemannian bounds. Likewise, we get a Lagrangian generalization of results of Hofer [19] and Viterbo [42] on simultaneous $C^0$ and Hofer/spectral limits — even without any such bounds.

We characterize the vanishing of the shifted Courant-Nijenhuis torsion as the strongest tensorial integrability condition that can be imposed on a skew-symmetric endomorphism of the generalized tangent bundle.

In this paper, we define and characterize 3-Sasakian statistical manifolds and then investigate statistical submersions from 3-Sasakian statistical manifolds. We prove that invariant statistical submersions from 3-Sasakian statistical manifolds with vertical structure vector fields have 3-Sasakian statistical totally geodesic fibers. Moreover, the base space admits a quaternionic Kähler statistical structure. We construct non-trivial examples to illustrate some results of the paper.

We present a review of known models and a new simple mathematical modelling for border completion in the visual cortex V1 highlighting the striking analogies with bicycle rear wheel motions in the plane.

Since the advent of new pairwise non-diffeomorphic structures on smooth manifolds, it has been questioned whether two topologically identical manifolds could admit different geometries. Not surprisingly, physicists have wondered whether a different smooth structure assumption to some classical known model could produce different physical meanings. Motivated by the works [27], [2], [3], [18], in this paper, we inaugurate a very computational manner to produce physical models on classical and exotic spheres that can be built equivariantly, such as the classical Gromoll–Meyer exotic spheres. As first applications, we produce Lorentzian metrics on homeomorphic but not diffeomorphic manifolds that enjoy the same physical properties, such as geodesic completeness, positive Ricci curvature, and compatible time orientation. These constructions can be pulled back to higher models, such as exotic ten spheres bounding spin manifolds, to be approached in forthcoming papers.

In this paper, we'll introduce the concept of sympathetic Lie conformal superalgebras and show that some classical properties of Lie conformal superalgebras are still valid for sympathetic Lie conformal superalgebras. We prove that the unique decomposition of each sympathetic Lie conformal superalgebra into a direct sum of indecomposable sympathetic ideals. We also show the existence of a greatest sympathetic ideal and a sympathetic decomposition in every perfect Lie conformal superalgebra. In the end, we also study the ideal $I$ of a Lie conformal superalgebra $R$ such that $R/I$ is a sympathetic Lie conformal superalgebra.