## Journal of the Mathematical Society of Japan

### Higher rank curved Lie triples

Jost-Hinrich ESCHENBURG

#### Abstract

A substantial proper submanifold $M$ of a Riemannian symmetric space $S$ is called a curved Lie triple if its tangent space at every point is invariant under the curvature tensor of $S$, i.e. a sub-Lie triple. E.g. any complex submanifold of complex projective space has this property. However, if the tangent Lie triple is irreducible and of higher rank, we show a certain rigidity using the holonomy theorem of Berger and Simons: $M$ must be intrinsically locally symmetric. In fact we conjecture that $M$ is an extrinsically symmetric isotropy orbit. We are able to prove this conjecture provided that a tangent space of $M$ is also a tangent space of such an orbit.

#### Article information

Source
J. Math. Soc. Japan, Volume 54, Number 3 (2002), 551-564.

Dates
First available in Project Euclid: 5 October 2007

https://projecteuclid.org/euclid.jmsj/1191593908

Digital Object Identifier
doi:10.2969/jmsj/1191593908

Mathematical Reviews number (MathSciNet)
MR1900956

Zentralblatt MATH identifier
1066.53099

#### Citation

ESCHENBURG, Jost-Hinrich. Higher rank curved Lie triples. J. Math. Soc. Japan 54 (2002), no. 3, 551--564. doi:10.2969/jmsj/1191593908. https://projecteuclid.org/euclid.jmsj/1191593908