The pseudoconvex and disprisoning conditions for geodesics of linear connections are extended to the solution curves of general homogeneous sprays. The main result is that pseudoconvexity and disprisonment are jointly stable in the fine topology on the space of all homogeneous sprays of any degree of homogeneity.
We survey recent results on sprays and give some examples of planar sprays.
The main purposes of this article are to extend our previous results on homogeneous sprays to arbitrary (generalized) sprays, to show that locally diffeomorphic exponential maps can be defined for any (generalized) spray, and to give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. In the process, we introduce vertically homogeneous connections. Unlike homogeneous connections, these allow us to include Finsler spaces among the applications.
We provide significant support for the prospect of studying nonlinear connections via (generalized) sprays. One of the most important is our generalized APS correspondence.
We show that locally diffeomorphic exponential maps can be defined for any second-order differential equation, and give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. We introduce vertically homogeneous connections as the natural correspondents of homogeneous second-order differential equations.
We provide significant support for the prospect of studying nonlinear connections via certain, closely associated second-order differential equations. One of the most important is our generalized Ambrose-Palais-Singer correspondence.
The main purpose of this article is to introduce a comprehensive, unified theory of the geometry of all connections. We show that one can study any connection via a certain, closely associated second-order differential equation, its geodesic quasispray. One of the most important results is our extended Ambrose-Palais-Singer correspondence. We extend the theory of geodesic sprays to the quasisprays, show that locally diffeomorphic exponential maps can be defined for any SODE, and give a full theory of (possibly nonlinear) covariant derivatives for (possibly nonlinear) connections. In the process, we introduce vertically homogeneous connections. Unlike homogeneous connections, these complete our theory and allow us to include Finsler spaces among the applications.
This is an expanded version of the article published in Differ. Geom. Dyn. Syst. 13 (2011) 72--90. Included are the proof published in Nonlinear Anal. 63 (2005) e501--e510 and some new material on homogeneity.
We introduce Jacobi fields for an SODE and use them to study its conjugate points and Jacobi stability of its geodesics. When a connection induces the SODE we recover the standard Jacobi equation, operator, and concomitants. We define general parallelism, and study parallel transport and holonomy of Ehresmann connections.
We study automorphism and holonomy groups of general and Ehresmann connections as locally convex Lie groups.
In the real plane, this is given by
On each of the following pages, the pictures show the geodesics, the concentric "circles" at "radii" (values of t , the geodesic parameter) 1,2,...,10, and the geodesics from -10 to 10 with initial velocities equally spaced around a circle, with height given by the value of t .
One of the implications of this is that the exponential image of straight lines through the origin in the tangent space need no longer be geodesics. For the spray with potential just preceeding, the result in three initial directions is this .
Another example of an inhomogeneous spray is given by
![[x'' = -y' and y'' = x]](sprays/inheqn.gif)
and a similar procedure yields this .