Variational problems on contact Riemannian manifolds

Author:
Shukichi Tanno

Journal:
Trans. Amer. Math. Soc. **314** (1989), 349-379

MSC:
Primary 53C15; Secondary 32F25, 58G30

DOI:
https://doi.org/10.1090/S0002-9947-1989-1000553-9

MathSciNet review:
1000553

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable manifolds. Then the torsion and the generalized Tanaka-Webster 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.

**[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), 82-88. MR**742245 (85f:58027)****[3]**S. S. Chern and R. S. Hamilton,*On Riemannian metrics adapted to three-dimensional contact manifolds*, Lecture Notes in Math., vol. 1111, Springer, Berlin, 1985, pp. 279-305. 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. 57-63. MR**741039 (85i:58122)****[5]**-,*The Yamabe problem on**manifolds*, J. Differential Geometry**25**(1987), 167-197. 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), 459-476. 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), 281-294. MR**0138065 (25:1513)****[9]**N. Tanaka,*On the pseudo-conformal geometry of hypersurfaces of the space of**complex variables*, J. Math. Soc. Japan**14**(1962), 387-429. MR**0145555 (26:3086)****[10]**-,*A differential geometric study on strongly pseudoconvex manifolds*, Lectures in Math., vol. 9, Kyoto Univ., 1975.**[11]**-,*On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections*, Japan. J. Math.**2**(1976), 131-190. MR**0589931 (58:28645)****[12]**S. Tanno,*Harmonic forms and Betti numbers of certain contact Riemannian manifolds*, J. Math. Soc. Japan**19**(1967), 308-316. MR**0212738 (35:3604)****[13]**-,*The topology of contact Riemannian manifolds*, Illinois J. Math.**12**(1968), 700-714. MR**0234486 (38:2803)****[14]**-,*The first eigenvalue of the Laplacian on spheres*, Tôhoku Math. J.**31**(1979), 179-185. MR**538918 (80g:58050)****[15]**-,*Some metrics on a*-*sphere and spectra*, Tsukuba J. Math.**4**(1980), 99-105. MR**597687 (82d:53028)****[16]**-,*Geometric expressions of eigen*-*forms of the Laplacian on spheres*, Spectra of Riemannian Manifolds, Kaigai, Tokyo, 1983, pp. 115-128.**[17]**S. M. Webster,*On the pseudo-conformal geometry of a Kähler manifold*, Math. Z.**157**(1977), 265-270. MR**0477122 (57:16666)****[18]**-,*Pseudo-Hermitian structure on a real hypersurface*, J. Differential Geometry**13**(1978), 25-41. MR**520599 (80e:32015)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53C15,
32F25,
58G30

Retrieve articles in all journals with MSC: 53C15, 32F25, 58G30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-1000553-9

Keywords:
Contact structure,
Tanaka-Webster scalar curvature,
gauge transformation of contact Riemannian structure

Article copyright:
© Copyright 1989
American Mathematical Society