Geometry of manifolds which admit conservation laws
Annales de l'Institut Fourier, Tome 21 (1971) no. 1, pp. 1-9.

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.

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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},
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     year = {1971},
     doi = {10.5802/aif.359},
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     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/

[1] D.E. Blair and A.P. Stone, A note on the holonomy group of manifolds with certain structures, Proc. AMS, 21 (1), (1969), 73-76. | MR 38 #5133 | Zbl 0175.48802

[2] A. Frölicher and A. Nijenhuis, Theory of vector valued differential forma, I ; Ned. Akad. Wet. Proc. 59 (1956), 338-359. | Zbl 0079.37502

[3] E.T. Kobayashi, A remark on the Nijenhuis tensor, Pacific J. Math., 12, (1962), 963-977. | MR 27 #678 | Zbl 0126.17901

[4] H. Osborn, The existence of conservation laws, I ; Ann. of Math., 69 (1959), 105-118. | MR 21 #760 | Zbl 0119.07801

[5] H. Osborn, Les lois de conservation, Ann. Inst. Fourier, (Grenoble), 14 (1964), 71-82. | Numdam | MR 30 #2425 | Zbl 0126.10904

[6] A.P. Stone, Analytic conservation laws, Ann. Inst. Fourier, (Grenoble), 16 (2), (1966), 319-327. | Numdam | MR 35 #6160 | Zbl 0168.07301

[7] A.P. Stone, Generalized conservation laws, Proc. AMS 18, (5), (1967), 868-873. | MR 36 #805 | Zbl 0159.13602

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