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Register a new userGeodesic Connectedness of Affine Manifolds
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We discuss new sufficient conditions under which an affine manifold $(M, abla)$ is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexander and Karr, with the added advantage that they yield an elementary proof of the main result.
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We describe all affine maps from a Riemannian manifold to a metric space and all possible image spaces.
First we confirm a conjecture asserting that any compact Kahler manifold $N$ with $Ric^perp>0$ must be simply-connected by applying a new viscosity consideration to Whitneys comass of $(p, 0)$-forms. Secondly we prove the projectivity and the rational connectedness of a Kahler manifold of complex dimension $n$ under the condition $Ric_k>0$ (for some $kin {1, cdots, n}$, with $Ric_n$ being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollar-Miyaoka-Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of cite{Ni-Zheng2}. Thirdly, motivated by $Ric^perp$ and the classical work of Calabi-Vesentini cite{CV}, we propose two new curvature notions. The cohomology vanishing $H^q(N, TN)={0}$ for any $1le qle n$ and a deformation rigidity result are obtained under these new curvature conditions. In particular they are verified for all classical Kahler C-spaces with $b_2=1$. The new conditions provide viable candidates for a curvature characterization of homogenous Kahler manifolds related to a generalized Hartshone conjecture.
Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $xin M$ and $TM={(x,y);xin M;y in T_xM}$ be its tangent bundle. A $C^0$-Finsler structure is a continuous function $F:TM rightarrow mathbb [0,infty)$ such that $F(x,cdot): T_xM rightarrow [0,infty)$ is an asymmetric norm. In this work we introduce the Pontryagin type $C^0$-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagins maximum principle for the problem of minimizing paths. We define the extended geodesic field $mathcal E$ on the slit cotangent bundle $T^ast Mbackslash 0$ of $(M,F)$, which is a generalization of the geodesic spray of Finsler geometry. We study the case where $mathcal E$ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by $mathcal E$ than by a similar structure on $TM$. Finally we show that the maximum of independent Finsler structures is a Pontryagin type $C^0$-Finsler structure where $mathcal E$ is a locally Lipschitz vector field.
We define systems of pre-extremals for the energy functional of regular rheonomic Lagrange manifolds and show how they induce well-defined Hamilton orthogonal nets. Such nets have applications in the modelling of e.g. wildfire spread under time- and space-dependent conditions. The time function inherited from such a Hamilton net induces in turn a time-independent Finsler metric - we call it the associated frozen metric. It is simply obtained by inserting the time function from the net into the given Lagrangean. The energy pre-extremals then become ordinary Finsler geodesics of the frozen metric and the Hamilton orthogonality property is preserved during the freeze. We compare our results with previous findings of G. W. Richards concerning his application of Huyghens principle to establish the PDE system for Hamilton orthogonal nets in 2D Randers spaces and also concerning his explicit spray solutions for time-only dependent Randers spaces. We analyze examples of time-dependent 2D Randers spaces with simple, yet non-trivial, Zermelo data; we obtain analytic and numerical solutions to their respective energy pre-extremal equations; and we display details of the resulting (frozen) Hamilton orthogonal nets.
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.
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