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Differentiable Manifolds with an Area Measure

Published online by Cambridge University Press:  20 November 2018

F. Brickell
F. Brickell*
Affiliation:
Northwestern University, Evanston, Illinois and Southampton University, Southampton, England
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In this section we fix some notations and give a definition of an area measure on a differentiate manifold, where throughout the paper the word differentiable implies differentiability of class C∞. Let M denote a differentiate manifold of dimension n and call a set of m linearly independent vectors {e1, … , em } at a point of M an m-frame of M. The set E′ of all such m-frames can be given the structure of a differentiable fibre bundle over M and we denote the projection of E' onto M by π′.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 540 - 549
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Brickell, F., On areal spaces, Tensor (N.S.), 13 (1963), 19–30.Google Scholar
2. Caratheodory, C., Über die Variationsrechnung bei merfachen Integralen, Acta Szeged, 4 (1929), 193–216.Google Scholar
3. Cartan, E., Les espaces de Finsler, Actualités Sci. Indust., No. 79 (Paris, 1934).Google Scholar
4. Cartan, E., Les espaces metriques fondés sur la notion d'aire, Actualités Sci. Ind., No. 72 (Paris, 1933).Google Scholar
5. Chern, S., On euclidean connections in a Finsler space, Proc. Nat. Acad. Sci. U.S.A., 29 (1943), 33–37.Google Scholar
6. Dedecker, P., Calcul des variations, formes différentielles et champs géodésiques, Coll. Int. de Géométrie Différentielle (Strasbourg, 1953).Google Scholar
7. Hocking, J. G. and Young, G. S., Topology, (Reading, Mass. and London, 1961).Google Scholar
8. Iwamoto, H., On geometries associated with multiple integrals, Math. Japon., I (1948), 74–91.Google Scholar
9. Kobayashi, S., Le groupe de transformations qui laissent invariant un parallélisme, Colloque de Topologie (Strasbourg, 1954).Google Scholar
10. Kobayashi, S. and Nomizu, K., Foundations of differential geometry, vol. I (New York and London, 1963).Google Scholar
11. Lichnerowicz, A., Quelques théorèmes de géométrie différentielle globale, Comment. Math. Helv., 22 (1949), 271–301.Google Scholar
12. Milnor, J., Morse theory, Ann. of Math. Studies, No. 51 (Princeton, 1963).Google Scholar
13. Palais, R. S., On the existence of slices for actions of non-compact Lie groups, Ann. Math., 73 (1961), 295–323.Google Scholar
14. Tandai, K., On areal spaces VI, Tensor (N.S.), 3 (1953), 40–45.Google Scholar
15. Tashiro, Y., A theory of transformation groups on generalised spaces and its applications to Finsler and Cartan spaces, J. Math. Soc. Japan, 11 (1959), 42–71.Google Scholar
16. Wang, H. C.. On Finsler spaces with completely integrable equations of Killing, J. London Math. Soc., 22(1947), 5–9.Google Scholar
17. Yano, K., On n-dimensional Riemannian spaces admitting a group of motions of order 1/2n(n-1) + 1, Trans. Amer. Math. Soc., 74 (1953), 260–279.Google Scholar