Geometry of manifolds which admit conservation laws

Blair, David E.; Stone, Alexander P.

doi:10.5802/aif.359

Blair, David E. ; Stone, Alexander P.

Annales de l'Institut Fourier, Tome 21 (1971) no. 1, pp. 1-9.

Résumé
Abstract

Soit $M$ une variété riemannienne à $(n + 1)$ dimensions, admettant un endomorphisme covariant constant $h$ du module local de 1-formes ayant des valeurs propres distinctes et différentes de zéro. On montre que $M$ est localement plat, et on étudie une variété $N$ immergée dans $M$ . La variété $N$ a une structure induite avec $n$ des mêmes valeurs propres si et seulement si la normale à $N$ est une direction fixe de $h$ . Enfin, on trouve les conditions sous lesquelles $N$ est invariant sous $h$ , $N$ est totalement géodésique et la structure induite a une torsion de Nijenhuis nulle ou est covariante constante.

Let $M$ be an $(n + 1)$ -dimensional Riemannian manifold admitting a covariant constant endomorphism $h$ of the localized module of 1-forms with distinct non-zero eigenvalues. After it is shown that $M$ is locally flat, a manifold $N$ immersed in $M$ is studied. The manifold $N$ has an induced structure with $n$ of the same eigenvalues if and only if the normal to $N$ is a fixed direction of $h$ . Finally conditions under which $N$ is invariant under $h$ , $N$ is totally geodesic and the induced structure has vanishing Nijenhuis torsion or is covariant constant are found.

MR Zbl

DOI : 10.5802/aif.359

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     author = {Blair, David E. and Stone, Alexander P.},
     title = {Geometry of manifolds which admit conservation laws},
     journal = {Annales de l'Institut Fourier},
     pages = {1--9},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {21},
     number = {1},
     year = {1971},
     doi = {10.5802/aif.359},
     mrnumber = {44 #948},
     zbl = {0197.18101},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.359/}
}

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Blair, David E.; Stone, Alexander P. Geometry of manifolds which admit conservation laws. Annales de l'Institut Fourier, Tome 21 (1971) no. 1, pp. 1-9. doi : 10.5802/aif.359. http://archive.numdam.org/articles/10.5802/aif.359/

Bibliographie
Cité par

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