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**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.”

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

### MSC:

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

### Keywords:

Tanaka-Webster scalar curvature; contact form; variational problems; contact manifold; strongly pseudoconvex integrable CR structure; torsion; gauge transformations
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\textit{S. Tanno}, Trans. Am. Math. Soc. 314, No. 1, 349--379 (1989; Zbl 0677.53043)

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### References:

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