Geometry of manifolds which admit conservation laws

Blair, David E.; Stone, Alexander P.

Annales de l'Institut Fourier, Tome 21 (1971), p. 1-9 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 1971-01-01

MR 44 #948 | Zbl 0197.18101

DOI : https://doi.org/10.5802/aif.359

@article{AIF_1971__21_1_1_0,
     author = {Blair, David E. and Stone, Alexander P.},
     title = {Geometry of manifolds which admit conservation laws},
     journal = {Annales de l'Institut Fourier},
     volume = {21},
     year = {1971},
     pages = {1-9},
     doi = {10.5802/aif.359},
     mrnumber = {44 \#948},
     zbl = {0197.18101},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1971__21_1_1_0}
}

Blair, David E.; Stone, Alexander P. Geometry of manifolds which admit conservation laws. Annales de l'Institut Fourier, Tome 21 (1971) pp. 1-9. doi : 10.5802/aif.359. http://gdmltest.u-ga.fr/item/AIF_1971__21_1_1_0/

Bibliographie

[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