Fixed points of isometries

@article{Kobayashi1958FixedPO,
  title={Fixed points of isometries},
  author={Sh{\^o}shichi Kobayashi},
  journal={Nagoya Mathematical Journal},
  year={1958},
  volume={13},
  pages={63-68}
}
The purpose of this paper is to prove the following Theorem. Let M be a Riemannian manifold of dimension n and let ξ be a Killing vector field (i.e., infinitesimal isometry) of M. Let F be the set of points x of M where ξ vanishes and let F = ∪ V i , where the V i ’s are the connected components of F. Then (assuming F to be non-empty ) 
Periodic orbits of isometric flows
Let M be a compact C ∞ Riemannian manifold, X a Killing vector field on M , and φ t its 1-parameter group of isometries of M . In this, paper, we obtain some basic properties of the set of periodicExpand
Singular Riemannian flows and characteristic numbers
Let M be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on M to a small tubular neighborhood of each connected component of its singularExpand
Some remarks on R-contact flows
Let (M, α) be an R-contact manifold, then the set of periodic points of the characteristic vector field is a nonempty union of closed, totally geodesic odd-dimensional submanifolds. Moreover, theExpand
ZERO POINTS OF KILLING VECTOR FIELDS, GEODESIC ORBITS, CURVATURE, AND CUT LOCUS
Let (A/, g) be a compact, connected, Riemannian manifold. Let X be a Killing vectot field on Ai. / = g(X, X) is called the length function of X. Let D denote the minimum of the distances from pointsExpand
INTRODUCTION TO SINGULAR RIEMANNIAN FOLIATIONS
A (smooth) foliation F of a smooth manifold M is a partition of M complete, connected, immersed submanifolds (leaves) of the same dimension such that for all x ∈M , there exists a distinguishedExpand
Conjugate loci of totally geodesic submanifolds of symmetric spaces
The conjugate and cut loci of fixed point sets of involutions which fix the origin of a compact symmetric space are studied. The first conjugate locus is described in terms of roots and weights ofExpand
The Group of Isometries
In this chapter we deal with isometries of a Finsler space. In the first two sections, we generalize the Myers-Steenrod theorem from Riemannian geometry to the Finslerian case, through a thoroughExpand
Bach-Flat Kähler Surfaces
A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the $$L^2$$ L 2 -norm of the Weyl curvature. When the Riemannian 4-manifold in question is a KählerExpand
Regular and quasiregular isometric flows on Riemannian manifolds
We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite)Expand
Smooth toral actions on principal bundles and characteristic classes
The purpose of our work is to find explicit formulae for the computation of some characteristic classes of smooth principal bundles P: P --> B, in terms of local invariants at a singular subset AG ofExpand
...
1
2
3
4
5
...