Circles in compact homogeneous Riemannian spaces and immersions of finite type
Glasgow Mathematical Journal
A unit speed curve ¼ ðsÞ in a Riemannian manifold N is called a circle if there exists a unit vector field YðsÞ along and a positive constant k such that r s 0 ðsÞ ¼ kYðsÞ; r s YðsÞ ¼ Àk 0 ðsÞ. The main purpose of this article is to investigate the fundamental relationships between circles, maximal tori in compact symmetric spaces, and immersions of finite type. 2000 Mathematics Subject Classification. Primary 53C35, 53C40. Secondary 53C30. 1. Introduction. A unit speed curve ¼ ðsÞ in a
... ¼ ðsÞ in a Riemannian manifold N is called a circle if there exists a vector field YðsÞ of unit vectors along and a positive constant k such that r s XðsÞ ¼ kYðsÞ; r s YðsÞ ¼ ÀkXðsÞ; where XðsÞ denotes the unit tangent vector of and r s the covariant differentiation along at each point ðsÞ. The number 1=k is called the radius of the circle . Let M be a Riemannian manifold. All isometries IðMÞ of M form a Lie group. Let G M be the connected component of IðMÞ that is compact if M is compact. A Riemannian manifold M is called homogeneous if G M acts transitively on M. Denote by K M the isotropy subgroup at a point o in M. We have M ¼ G M =K M . Very often, we simply denote G M and K M by G and K, respectively. A curve in M ¼ G M =K M is called a homogenous curve if it is the orbit of a point under the action of a one-parameter subgroup f t g of G M . The linear isotropy representation of M is the orthogonal representation of K over the tangent space T o M at o defined by K ! OðT o MÞ : 7 !ð Ã Þ o ; where ð Ã Þ o denotes the differential of the isometry at o. A homogeneous Riemannian manifold is said to be isotropyirreducible if its linear isotropy representation is irreducible. Let ; h i denote the Ad G ðKÞ-invariant inner product on M ¼ G=K. Let g and k be the Lie algebras of G and K, and g ¼ k È m the Cartan decomposition of g. M is called naturally reductive if ½Z; X m ; Y þ X; ½Z; Y m ¼ 0; for X; Y; Z 2 m, where ½ ; m is the m-component of the Lie bracket in g. The purpose of this article is to investigate the fundamental relationships between circles, maximal tori in compact symmetric spaces, and immersions of finite type. Our main results are the following. (a) An isometric immersion of a compact irreducible symmetric space M into Euclidean space is of finite type if and only if it carries maximal tori of M into submanifolds of finite type.