Variational problems on contact Riemannian manifolds
Author:
Shukichi Tanno
Journal:
Trans. Amer. Math. Soc. 314 (1989), 349379
MSC:
Primary 53C15; Secondary 32F25, 58G30
MathSciNet review:
1000553
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Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable manifolds. Then the torsion and the generalized TanakaWebster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.
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 [1]
 D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math., vol. 509, Springer, Berlin, 1976. MR 0467588 (57:7444)
 [2]
 , Critical associated metrics on contact manifolds, J. Austral. Math. Soc. 37 (1984), 8288. MR 742245 (85f:58027)
 [3]
 S. S. Chern and R. S. Hamilton, On Riemannian metrics adapted to threedimensional contact manifolds, Lecture Notes in Math., vol. 1111, Springer, Berlin, 1985, pp. 279305. MR 797427 (87b:53060)
 [4]
 D. Jerison and J. M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on manifolds, Contemp. Math., vol. 27, Amer. Math. Soc., Providence, R.I., 1984, pp. 5763. MR 741039 (85i:58122)
 [5]
 , The Yamabe problem on manifolds, J. Differential Geometry 25 (1987), 167197. MR 880182 (88i:58162)
 [6]
 , Extremals for the Sobolev inequality on the Heisenberg group and the Yamabe problem, preprint.
 [7]
 S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure. I, Tôhoku Math. J. 12 (1960), 459476. MR 0123263 (23:A591)
 [8]
 S. Sasaki and Y. Hatakeyama, On differentiable manifolds with certain structures which are closely related to almost contact structure. II, Tôhoku Math. J. 13 (1961), 281294. MR 0138065 (25:1513)
 [9]
 N. Tanaka, On the pseudoconformal geometry of hypersurfaces of the space of complex variables, J. Math. Soc. Japan 14 (1962), 387429. MR 0145555 (26:3086)
 [10]
 , A differential geometric study on strongly pseudoconvex manifolds, Lectures in Math., vol. 9, Kyoto Univ., 1975.
 [11]
 , On nondegenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. 2 (1976), 131190. MR 0589931 (58:28645)
 [12]
 S. Tanno, Harmonic forms and Betti numbers of certain contact Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 308316. MR 0212738 (35:3604)
 [13]
 , The topology of contact Riemannian manifolds, Illinois J. Math. 12 (1968), 700714. MR 0234486 (38:2803)
 [14]
 , The first eigenvalue of the Laplacian on spheres, Tôhoku Math. J. 31 (1979), 179185. MR 538918 (80g:58050)
 [15]
 , Some metrics on a sphere and spectra, Tsukuba J. Math. 4 (1980), 99105. MR 597687 (82d:53028)
 [16]
 , Geometric expressions of eigen forms of the Laplacian on spheres, Spectra of Riemannian Manifolds, Kaigai, Tokyo, 1983, pp. 115128.
 [17]
 S. M. Webster, On the pseudoconformal geometry of a Kähler manifold, Math. Z. 157 (1977), 265270. MR 0477122 (57:16666)
 [18]
 , PseudoHermitian structure on a real hypersurface, J. Differential Geometry 13 (1978), 2541. MR 520599 (80e:32015)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198910005539
PII:
S 00029947(1989)10005539
Keywords:
Contact structure,
TanakaWebster scalar curvature,
gauge transformation of contact Riemannian structure
Article copyright:
© Copyright 1989
American Mathematical Society
