Critical mappings of Riemannian manifolds

Bleecker, David D.

doi:10.1090/S0002-9947-1979-0539921-1

Critical mappings of Riemannian manifolds
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by David D. Bleecker PDF

Trans. Amer. Math. Soc. 254 (1979), 319-338 Request permission

Abstract:

We consider maps, from one Riemannian manifold to another, which are critical for all invariantly defined functionals on the space of maps. There are many such critical mappings, perhaps too numerous to suitably classify, although a characterization of sorts is provided. They are proven to have constant rank, with the image being a homogeneous minimal submanifold of the target manifold. Critical maps need not be Riemannian submersions onto their images. Also, there are homogeneous spaces for which the identity map is not critical. Many open problems remain.

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