# Geometry of manifolds which admit conservation laws

David E. Blair; Alexander P. Stone

Annales de l'institut Fourier (1971)

- Volume: 21, Issue: 1, page 1-9
- ISSN: 0373-0956

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topBlair, David E., and Stone, Alexander P.. "Geometry of manifolds which admit conservation laws." Annales de l'institut Fourier 21.1 (1971): 1-9. <http://eudml.org/doc/74026>.

@article{Blair1971,

abstract = {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.},

author = {Blair, David E., Stone, Alexander P.},

journal = {Annales de l'institut Fourier},

keywords = {differential geometry},

language = {eng},

number = {1},

pages = {1-9},

publisher = {Association des Annales de l'Institut Fourier},

title = {Geometry of manifolds which admit conservation laws},

url = {http://eudml.org/doc/74026},

volume = {21},

year = {1971},

}

TY - JOUR

AU - Blair, David E.

AU - Stone, Alexander P.

TI - Geometry of manifolds which admit conservation laws

JO - Annales de l'institut Fourier

PY - 1971

PB - Association des Annales de l'Institut Fourier

VL - 21

IS - 1

SP - 1

EP - 9

AB - 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.

LA - eng

KW - differential geometry

UR - http://eudml.org/doc/74026

ER -

## References

top- [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. Zbl0175.48802MR38 #5133
- [2] A. FRÖLICHER and A. NIJENHUIS, Theory of vector valued differential forma, I ; Ned. Akad. Wet. Proc. 59 (1956), 338-359. Zbl0079.37502
- [3] E.T. KOBAYASHI, A remark on the Nijenhuis tensor, Pacific J. Math., 12, (1962), 963-977. Zbl0126.17901MR27 #678
- [4] H. OSBORN, The existence of conservation laws, I ; Ann. of Math., 69 (1959), 105-118. Zbl0119.07801MR21 #760
- [5] H. OSBORN, Les lois de conservation, Ann. Inst. Fourier, (Grenoble), 14 (1964), 71-82. Zbl0126.10904MR30 #2425
- [6] A.P. STONE, Analytic conservation laws, Ann. Inst. Fourier, (Grenoble), 16 (2), (1966), 319-327. Zbl0168.07301MR35 #6160
- [7] A.P. STONE, Generalized conservation laws, Proc. AMS 18, (5), (1967), 868-873. Zbl0159.13602MR36 #805

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