Differential Geometry and its Applications最新文献

The weighted Yamabe flow is the geometric flow introduced to study the weighted Yamabe problem on smooth metric measure spaces. Carlotto, Chodosh and Rubinstein have studied the convergence rate of the Yamabe flow. Inspired by their result, we study in this paper the convergence rate of the weighted Yamabe flow.

In this paper, we study weakly stretch Kropina metrics and prove a rigidity theorem. We show that the associated one-form of a weakly stretch Kropina metric is conformally Killing with respect to the associated Riemannian metric. We find that any weakly stretch Kropina metric has vanishing S-curvature. Then, we prove that every weakly stretch Kropina metric is a Berwald metric. It turns out that every R-quadratic Kropina metric is a Berwald metric. Finally, we show that every positively complete C-reducible metric is R-quadratic if and only if it is Berwaldian.

In this paper, we study Finsler warped product metrics. We obtain the differential equations that characterize Landsberg Finsler warped product metrics. By solving these equations, we obtain the expression of these metrics. Furthermore, we construct a class of almost regular Finsler warped product metrics F with the following properties: (1) F is a Landsberg metric; (2) F is not a Berwald metric; (3) F has zero flag curvature (or Ricci curvature).

In this paper we discuss the categorical properties of $Z$-graded manifolds. We start by describing the local model paying special attention to the differences in comparison to the $N$-graded case. In particular we explain the origin of formality for the functional space and spell-out the structure of the power series. Then we make this construction intrinsic using a new type of filtrations. This sums up to proper definitions of objects and morphisms in the category. We also formulate the analog of the Borel's lemma for the functional spaces on $Z$-graded manifolds and the analogue of Batchelor's theorem for the global structure of them.

We prove a structure theorem for homogeneous closed gradient Laplacian solitons and use it to show some examples of closed Laplacian solitons cannot be made gradient. More specifically, we show that the Laplacian solitons on nilpotent Lie groups found by Nicolini are not gradient up to homothetic $G_2$-structures except for $R^7$, where the potential function must be of a certain form. We also show that one of the closed $G_2$-structures constructed by Fernández-Fino-Manero cannot be a gradient soliton. We then examine the structure of almost abelian solvmanifolds admitting closed non-torsion-free gradient Laplacian solitons.

The class of warped product metrics can often be interpreted as key space models for general theory of relativity and in the theory of space-time structure. In this paper, we study one of the most important non-Riemannian quantities in Finsler geometry which is called the S-curvature. We examined the behavior of the S-curvature in the Finsler warped product metrics. We are going to prove that every Finsler warped product metric $R\times R^n$ has almost isotropic S-curvature if and only if it is a weakly Berwald metric. Moreover, we show that every Finsler warped product metric has isotropic S-curvature if and only if S-curvature vanishes.

An important open question related to the study of $G_2$-holonomy manifolds concerns whether or not a compact seven-manifold can support an exact $G_2$-structure. To provide insight into this question, we identify various relationships between the two-form underlying an exact $G_2$-structure, the torsion of the $G_2$-structure, and the curvatures of the associated metric. In addition to establishing identities valid for any hypothetical exact $G_2$-structure, we also consider exact $G_2$-structures subject to additional constraints, for instance proving incompatibility between the exact $G_2$ and Extremally Ricci-Pinched conditions and establish new identities for soliton solutions of the Laplacian flow.

Working in the framework of nilpotent geometry, we give a unified scheme for the equivalence problem of geometric structures which extends and integrates the earlier works by Cartan, Singer-Sternberg, Tanaka, and Morimoto.

By giving a new formulation of the higher order geometric structures and the universal frame bundles, we reconstruct the step prolongation of Singer-Sternberg and Tanaka. We then investigate the structure function γ of the complete step prolongation of a proper geometric structure by expanding it into components $\gamma =\kappa +\tau +\sigma$ and establish the fundamental identities for κ, τ, σ. This then enables us to study the equivalence problem of geometric structures in full generality and to extend applications largely to the geometric structures which have not necessarily Cartan connections.

Among all we give an algorithm to construct a complete system of invariants for any higher order proper geometric structure of constant symbol by making use of generalized Spencer cohomology group associated to the symbol of the geometric structure. We then discuss thoroughly the equivalence problem for geometric structure in both cases of infinite and finite type.

We also give a characterization of the Cartan connections by means of the structure function τ and make clear where the Cartan connections are placed in the perspective of the step prolongations.

In this paper, we discuss how to travel along horizontal broken geodesics of a homogeneous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets $A_q\left(C\right)$ of the set of analytic vector fields $C$ determined by the family of horizontal unit geodesic vector fields $C$ to the fibers $F=\left\{{\rho }^{-1}\right(c\left)\right\}$ of a homogeneous analytic Finsler submersion $\rho :M\to B$. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examples on non compact Finsler manifolds M where the attainable sets (the dual leaves) are no longer orbits or even submanifolds. Nevertheless we prove that, when M is compact and the orbits of $C$ are embedded, then the attainable sets coincide with the orbits. Furthermore, if the flag curvature is positive then M coincides with the attainable set of each point. In other words, fixed two points of M, one can travel from one point to the other along horizontal broken geodesics.

In addition, we show that each orbit $O\left(q\right)$ of $C$ associated to a singular Finsler foliation coincides with M, when the flag curvature is positive, i.e., we prove Wilking's result in Finsler context. In particular we review Wilking's transversal Jacobi fields in Finsler case.

We characterize Lagrangian submanifolds in complex projective space for which each parallel submanifold along normal geodesics with respect to a unit normal vector field is Lagrangian, by using a normal line congruence of the Lagrangian submanifold to complex 2-plane Grassmannian and quaternionic Kähler structure. As a special case, we can construct minimal ruled Lagrangian submanifolds in complex projective space from an austere hypersurface in sphere with non-vanishing Gauss-Kronecker curvature.