Journal of the Mathematical Society of Japan

Higher rank curved Lie triples


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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

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

First available in Project Euclid: 5 October 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C35: Symmetric spaces [See also 32M15, 57T15]
Secondary: 53C24: Rigidity results 53C29: Issues of holonomy 53B25: Local submanifolds [See also 53C40]

Symmetric spaces restricted holonomy submanifold equations Gauss map rank rigidity extrinsic symmetry isotropy orbits


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

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