Differential Geometry and its Applications最新文献

A smooth manifold hosts different types of submanifolds, including embedded, weakly-embedded, and immersed submanifolds. The notion of an immersed submanifold requires additional structure (namely, the choice of a topology); when this additional structure is unique, we call the subset a uniquely immersed submanifold. Diffeology provides yet another intrinsic notion of submanifold: a diffeological submanifold.

We show that from a categorical perspective diffeology rises above the others: viewing manifolds as a concrete category over the category of sets, the initial morphisms are exactly the (diffeological) inductions, which are the diffeomorphisms with diffeological submanifolds. Moreover, if we view manifolds as a concrete category over the category of topological spaces, we recover Joris and Preissmann's notion of pseudo-immersions.

We show that these notions are all different. In particular, a theorem of Joris from 1982 yields a diffeological submanifold whose inclusion is not an immersion, answering a question that was posed by Iglesias-Zemmour. We also characterize local inductions as those pseudo-immersions that are locally injective.

In appendices, we review a proof of Joris' theorem, pointing at a flaw in one of the several other proofs that occur in the literature, and we illustrate how submanifolds inherit paracompactness from their ambient manifold.

For an asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) gravitational instanton $(M,g)$ with toric symmetry, we express the signature of $(M,g)$ directly in terms of its rod structure. Applying Hitchin–Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, we formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. Finally, we apply these results to the study of rod structures with three turning points.

We prove Schwarz-type lemma results for Weierstrass parameterization of the minimal disk in the Riemannian manifold $M\times R$, where M is a Riemannian surface of non-positive Gaussian curvature. The estimate is sharp, and the equality is attained if and only if the ϱ-harmonic mapping that produces the parameterization is conformal and the metric is a Euclidean metric. If the area of the minimal surface is equal to the area of the disk, then the parametrization is a contraction w.r.t. induced metric and hyperbolic metric respectively.

We classify equivariant harmonic maps of the complex projective spaces $C{P}^m$ into the quaternion projective spaces. To do this, we employ differential geometry of vector bundles and connections. When the domain is the complex projective line, we have one parameter family of those maps. (This result is already shown in [2] and [4] in other ways). However, when $m\geqq 2$, we will obtain the rigidity results.

The gluing formula for the zeta-determinants of Laplacians with respect to the Robin boundary condition was proved in [15]. This formula contains a constant which is expressed by some curvature tensors on the cutting hypersurface including the scalar and principal curvatures. In this paper we compute this constant explicitly when the cutting hypersurface is a 2-dimensional closed submanifold in a closed Riemannian manifold, and discuss some related topics. We next use the conformal rescaling of the Riemannian metric to compute the value of the zeta function at zero associated to the generalized Dirichlet-to-Neumann operator defined by the Robin boundary condition on this cutting hypersurface.

Let $\Omega =G/K$ be a bounded symmetric domain of non-compact type. In this paper the image of the Poisson transform on the degenerate principal series representations of G attached to the Shilov boundary of Ω is considered. We characterize the images in terms of the third-order Hua operators $U_{\nu }$ and $W_{\nu }$. When Ω is the exceptional domain of type V, we give the explicit formulas for the operators $U_{\nu }$ and $W_{\nu }$.

In a previous paper [18] the authors employed the fiber join construction of Yamazaki [38] together with the admissible construction of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [2] to construct new extremal Sasaki metrics on odd dimensional sphere bundles over smooth projective algebraic varieties. In the present paper we continue this study by applying a recent existence theorem [14] that shows that under certain conditions one can always obtain a constant scalar curvature Sasaki metric in the Sasaki cone. Moreover, we explicitly describe this construction for certain sphere bundles of dimension 5 and 7.

In this paper, we deal with non-degenerate translators of the mean curvature flow in the well-known Einstein's static universe. We focus on the rotationally invariant translators, that is, those invariant by a natural isometric action of the special orthogonal group on the ambient space. In the classification list, there are three space-like cases and five time-like cases. All of them, except a totally geodesic example, have one or two conic singularities. Also, we show a uniqueness result based on the behaviour of the translator on its boundary. As an application, we extend an isometry of the sphere to the whole translator under simple conditions. This leads to a characterization of a bowl-like example.

We give a complete classification of isoparametric hypersurfaces in a product space $M_{{\kappa }_1}^2\times M_{{\kappa }_2}^2$ of 2-dimensional space forms for ${\kappa }_i\in \{-1,0,1\}$ with ${\kappa }_1\ne {\kappa }_2$. In fact we prove that any isoparametric hypersurface in such a space has constant product angle function, which enables us to remove the condition of constant principal curvatures from the classification obtained recently by J.B.M. dos Santos and J.P. dos Santos.

The present paper is devoted to the investigation of absolutely continuous curves in asymmetric metric spaces induced by Finsler structures. Firstly, for asymmetric spaces induced by Finsler manifolds, we show that three different kinds of absolutely continuous curves coincide when their domains are bounded closed intervals. As an application, a universal existence and regularity theorem for gradient flow is obtained in the Finsler setting. Secondly, we study absolutely continuous curves in Wasserstein spaces over Finsler manifolds and establish the Lisini structure theorem in this setting, which characterize the nature of absolutely continuous curves in Wasserstein spaces in terms of dynamical transference plans concentrated on absolutely continuous curves in base Finsler manifolds. Besides, a close relation between continuity equations and absolutely continuous curves in Wasserstein spaces is founded. Last but not least, we also consider nonsmooth “Finsler-like” spaces, in which case most of the aforementioned results remain valid. Various model examples are constructed in this paper, which point out genuine differences between the asymmetric and symmetric settings.