Totally Geodesic Orbits of Groups of Isometries
Totally Geodesic Orbits of Groups of Isometries
复制标题
等距群的全测地线轨道
DOI：
10.1016/s1385-7258(62)50027-9
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发表时间：
1962
期刊：
影响因子：
--
通讯作者：
R. Hermann
中科院分区：
文献类型：
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作者：
R. Hermann
我们的一般问题是：给定一个黎曼流形M的等距的连通李群L，L的哪些轨道在M中是全测地的？在L是一维的情况下，[4] 2）给出了一个或多或少完整的答案：M上存在一个函数，其临界点是测地轨道。考虑其他简单的几何例子表明，这种现象是更普遍的，全测地轨道在某种意义上是轨道空间或轨道空间上的函数的临界点3）。我们关于全测地子空间和轨道的工作将分为两部分：（a）关于紧致不可约对称空间上某些极大等距群的全测地轨道的描述。(b)Killing向量场长度函数临界点的Hessian性质和几何性质的进一步研究。所有流形、张量场、群和群的作用、映射等都是可微类coo，除非另有说明。假设所有的群作为变换群有效地起作用，即除了单位元之外，群中没有任何元素引起单位元变换。设M是流形，Mx表示M在x EM处的切空间，C（M）表示M上的coo实值函数环，V（M）表示M上的向量场集，它既被看作C（M）上的模，又被看作Jacobi括号运算下的真实的李代数。设G是作用在M上的李群。ForgE G和x EM，gx，g· x或Tg（x）将表示x通过g的变换。G，G的李代数，可以用V（M）的李子代数来标识：对于X E G，x EM，X（x）是切向量1）马萨诸塞州理工学院的林肯实验室，在美国陆军、海军和空军的支持下运作。2)文[4]将M中位于测地轨道上的点刻画为一个函数的临界点在M K切丛上的投影。N omizu指出，它们可以更简单地描述为L的无穷小生成元Killing向量场的长度的临界点。3)例如，考虑球面的旋转群使北极不变。此外，T。Frankel在[3]中证明了，在L是通过伴随作用于自身的经典群的情况下，全测地轨道是迹函数的临界点。
Our general problem is: Given a connected Lie group L of isometries of a Riemannian manifold M, which orbits of L are totally geodesic in M? In case L is one-dimensional, a more-or-less complete answer has been given in [4] 2): There is a function on M whose critical points are the geodesic orbits. Consideration of other simple geometric examples suggests that the phenomenon is more general, that the totally geodesic orbits are in some sense critical points of the orbit space or of a function on the orbit space 3). Our work on totally geodesic sub-spaces and orbits will be devided into two parts:(a) Description of the totally geodesic orbits of certain maximal groups of isometries on compact, irreducible symmetric spaces.(b) Further investigation of the Hessian and geometric properties of the critical points of the length function of a Killing vector field. All manifolds, tensor-fields, groups and action of groups, maps, etc. will be of differentiability class coo unless mentioned otherwise. It will be assumed that all groups acting as transformation groups act effectively, ie no element of the group except the identity induces the identity transformation. If M is a manifold, Mx denotes the tangent space to M at x EM, C (M) denotes the ring of coo real-valued functions on M, V (M) the set of vector-fields on M considered both as a module over C (M) and as a real Lie algebra under the Jacobi bracket operation. Let G be a Lie group acting on M. ForgE G and x EM, gx, g· x or Tg (x) will denote the transform of x by g. G, the Lie algebra of G, may be identified with a Lie subalgebra of V (M): For X E G, x EM, X (x) is the tangent vector1) Lincoln Laboratory, Massachusetts Institute of Technology, operated with support from the US Army, Navy, and Air Force. 2) In [4], the points of M lying on geodesic orbits were characterized as the projection of the critical points of a function on the tangent bundle of MK N omizu has pointed out that they can be more simply described as the critical point of the length of the Killing vector field that is the infinitesimal generator of L. 3) Think, for example, of the group of rotations of the sphere leaving the north pole invariant. Further, T. Frankel has shown [3] that, in the case where L is a classical group acting on itself via the adjoint action, the totally geodesic orbits are the critical points of the trace function.