Variational problems on contact Riemannian manifolds. (English) Zbl 0677.53043

From the introduction: “The motivation of this paper is the following: Since there are so many Riemannian metrics associated with a contact form \(\eta\) on a contact manifold (M,\(\eta)\) we want to find some proper one among them canonically related to \(\eta\). One method of finding conditions of nice Riemannian metrics is to study variational problems and their critical points. Furthermore, in the study of a contact manifold (M,\(\eta)\) it is desirable to find differential geometric properties which are independent of the choice of the contact form \(f\eta\), f being a positive function on M. So, we study gauge transformations of contact Riemannian structures.
One basic idea is to define the canonical connection, the torsion tensor and the scalar curvature for a contact Riemannian structure considering contact Riemannian structures as a generalization of strongly pseudoconvex integrable CR structure. Then we can study variational problems related to the torsion or scalar curvature and we obtain an invariant under gauge transformations of contact Riemannian structures.”
Reviewer: L.Ornea


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text: DOI


[1] David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. · Zbl 0319.53026
[2] David E. Blair, Critical associated metrics on contact manifolds, J. Austral. Math. Soc. Ser. A 37 (1984), no. 1, 82 – 88. · Zbl 0552.53014
[3] S. S. Chern and R. S. Hamilton, On Riemannian metrics adapted to three-dimensional contact manifolds, Workshop Bonn 1984 (Bonn, 1984) Lecture Notes in Math., vol. 1111, Springer, Berlin, 1985, pp. 279 – 308. With an appendix by Alan Weinstein.
[4] David Jerison and John M. Lee, A subelliptic, nonlinear eigenvalue problem and scalar curvature on CR manifolds, Microlocal analysis (Boulder, Colo., 1983) Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 57 – 63.
[5] David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167 – 197. · Zbl 0661.32026
[6] -, Extremals for the Sobolev inequality on the Heisenberg group and the \( CR\) Yamabe problem, preprint. · Zbl 0634.32016
[7] Shigeo Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure. I, Tôhoku Math. J. (2) 12 (1960), 459 – 476. · Zbl 0192.27903
[8] Shigeo Sasaki and Yoji Hatakeyama, On differentiable manifolds with certain structures which are closely related to almost contact structure. II, Tôhoku Math. J. (2) 13 (1961), 281 – 294. · Zbl 0112.14002
[9] Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of \? complex variables, J. Math. Soc. Japan 14 (1962), 397 – 429. · Zbl 0113.06303
[10] -, A differential geometric study on strongly pseudoconvex manifolds, Lectures in Math., vol. 9, Kyoto Univ., 1975.
[11] Noboru Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), no. 1, 131 – 190. · Zbl 0346.32010
[12] Shûkichi Tanno, Harmonic forms and Betti numbers of certain contact Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 308 – 316. · Zbl 0158.40202
[13] Shûkichi Tanno, The topology of contact Riemannian manifolds, Illinois J. Math. 12 (1968), 700 – 717. · Zbl 0165.24703
[14] Shûkichi Tanno, The first eigenvalue of the Laplacian on spheres, Tôhoku Math. J. (2) 31 (1979), no. 2, 179 – 185. · Zbl 0393.53024
[15] Shûkichi Tanno, Some metrics on a (4\?+3)-sphere and spectra, Tsukuba J. Math. 4 (1980), no. 1, 99 – 105. · Zbl 0458.53023
[16] -, Geometric expressions of eigen \( 1\)-forms of the Laplacian on spheres, Spectra of Riemannian Manifolds, Kaigai, Tokyo, 1983, pp. 115-128.
[17] Sidney M. Webster, On the pseudo-conformal geometry of a Kähler manifold, Math. Z. 157 (1977), no. 3, 265 – 270. · Zbl 0354.53022
[18] S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), no. 1, 25 – 41. · Zbl 0379.53016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.