Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Diffeomorphisms and volume-preserving embeddings of noncompact manifolds

Authors: R. E. Greene and K. Shiohama
Journal: Trans. Amer. Math. Soc. 255 (1979), 403-414
MSC: Primary 58D10; Secondary 53C42
MathSciNet review: 542888
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The theorem of J. Moser that any two volume elements of equal total volume on a compact manifold are diffeomorphism-equivalent is extended to noncompact manifolds: A necessary and sufficient condition (equal total and same end behavior) is given for diffeomorphism equivalence of two volume forms on a noncompact manifold. Results on the existence of embeddings and immersions with the property of inducing a given volume form are also given. Generalizations to nonorientable manifolds and manifolds with boundary are discussed.

References [Enhancements On Off] (What's this?)

  • [1] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trans. Moscow Math. Soc. 23 (1970), 1-35. MR 0370662 (51:6888)
  • [2] A. Banyaga, Formes-volume sur les variétés à bord, Enseignement Math. (2) 20 (1974), 127-131. MR 0358649 (50:11108)
  • [3] R. E. Greene and K. Shiohama, Volume-preserving diffeomorphisms and embeddings of noncompact manifolds. Notices Amer. Math. Soc. 23 (1976), Abstract A-446.
  • [4] M. L. Gromov, Smoothing and inversion of differential operators, Math. USSR-Sb. 17 (1972), 381-435. MR 0310924 (46:10022)
  • [5] M. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242-276. MR 0119214 (22:9980)
  • [6] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286-294. MR 0182927 (32:409)
  • [7] J. Nash, $ {C^1}$ isometric embeddings, Ann. of Math. (2) 60 (1954), 383-396. MR 0065993 (16:515e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58D10, 53C42

Retrieve articles in all journals with MSC: 58D10, 53C42

Additional Information

Keywords: Volume form, diffeomorphism, ends of a manifold, volume-preserving embedding, isometric embedding, odd forms
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society