A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.
We study the interplay between geodesics on two non-holono\-mic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.
In this paper, we introduce the associated geodesic-Einstein flow for a relatively ample line bundle $L$ over the total space $\mathcal{X}$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $(\mathcal{X}, L)$ is nonlinear semistable if the {associated} Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time {existence property}. We also define the associated $S$-classes and $C$-classes for $(\mathcal{X}, L)$ and obtain two inequalities between them when $L$ admits a geodesic-Einstein metric. Finally, …
Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit metric. We study the geodesic orbit spaces of the form $(\Sp(n)/\Sp(n_1)\times \cdots \times \Sp(n_s), g)$, with $0<n_1+\cdots +n_s\leq n$. Such spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces $(G/H,g)$ with $H$ semisimple.
Transverse totally geodesic submanifolds of the tangent bundle
众所周知，如果$ξ$是一个给定黎曼流形$M^n$上的光滑向量场，那么$ξ$自然定义了一个横截于切丛$TM^n$的纤维的子流形$ξ(M^n)$，且该子流形具有Sasaki度量.本文研究了切丛的横截全测地子流形。证明了$TM^n$（$1 \leq l \leq n$）的横子流形$N^l$可以局部地实现为$M^n$的子流形$F^l$在沿$F^l$定义的某个向量场$ξ$下的像.对于这样的图像$ξ(F^l)$，条件是完全测地线。我们证明了这些条件并不像$l=n$那样严格，并且我们处理了几种特殊情况（$ξ$的常长，$ξ$垂直于$F^l$，$M^n$的常曲率，$M^n$李群和$ξ$左不变向量场）
In this paper we study geodesics on adjoint orbits of $SL(n,\mathbb{R})$ equipped with $SO(n)$-invariant metrics (maximal compact subgroup). Our main technique is translate this problem into a geometric problem in the tangent bundle of certain $SO(n)$-flag manifolds and describe the geodesics equations with respect to the Sasaki metric on tangent bundle. We also use tools of Lie Theory in order to obtain some explicit description of families of geodesics. We deal with the case of $SL(2,\mathbb{R})$ in full details.
The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}, \] under the Carathéodory metric, is a complex Finsler space of cohomogeneity $1$ in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, $G$ does not admit a natural notion of angle, but we nevertheless show that there {\em is} a notion of orthogonality. The complex tangent bundle $TG$ splits naturally into the direct sum of two line bundles, which we call the {\em sharp} and {\em flat} bundles, and which are geometrically defined and therefore covariant under automorphisms of $G…
In the semiclassical limit h to 0, we analyze a class of self-adjoint Schrödinger operators H_h = h^2 L + h W + V id_E acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m_1, ... m_r in M, called potential wells. Using quasimodes of WKB-type near m_j for eigenfunctions associated with the low lying eigenvalues of H_h, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in cer…
Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $x\in M$ and $TM=\{(x,y);x\in M;y \in T_xM\}$ 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 Pontryagin's maximum principle for the problem of minimizing paths. We define the extended geodesic field $\mathcal E$ on the slit cotangent bundle $T^\ast M\backsl…
There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as (locally) shortest curves (Euler-Lagrange, Pontyagin and Hamilton) and three definitions of geodesics as straightest curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in nonholonomic mechanics and discuss their interrelations. We consider a big class of sub-Rie…
