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Transactions of the American Mathematical Society

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Nonexistence of nontrivial $ cm''$-harmonic 1-forms on a complete foliated Riemannian manifold


Author: Haruo Kitahara
Journal: Trans. Amer. Math. Soc. 262 (1980), 429-435
MSC: Primary 57R30; Secondary 58A14
MathSciNet review: 586726
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Abstract: We study the nonexistence of nontrivial $ \square''$-harmonic 1-forms on a complete foliated riemannian manifold with positive definite Ricci curvature. It is well known that the harmonic 1-form on a compact and orientable riemannian manifold with positive definite Ricci curvature is trivial. Our main theorem is an extension of this fact in the complete foliated riemannian case.


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

  • [1] Haruo Kitahara, Remarks on square-integrable basic cohomology spaces on a foliated Riemannian manifold, Kodai Math. J. 2 (1979), no. 2, 187–193. MR 548394
  • [2] Haruo Kitahara, On a parametrix form in a certain 𝑉-submersion, Geometry and differential geometry (Proc. Conf., Univ. Haifa, Haifa, 1979), Lecture Notes in Math., vol. 792, Springer, Berlin, 1980, pp. 264–298. MR 585877
  • [3] Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 0107279
  • [4] Bruce L. Reinhart, Harmonic integrals on foliated manifolds, Amer. J. Math. 81 (1959), 529–536. MR 0107280
  • [5] Izu Vaisman, Variétés riemanniennes feuilletées, Czechoslovak Math. J. 21 (96) (1971), 46–75 (French). MR 0287572

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0586726-X
Keywords: Bundle-like metric, second connection, $ \square''$-harmonic form
Article copyright: © Copyright 1980 American Mathematical Society