Geodesic
A Variational Characterization of Contact Metric Manifolds With Vanishing Torsion
Canadian mathematical bulletin, Tome 35 (1992) no. 4, pp. 455-462

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Chern and Hamilton considered the integral of the Webster scalar curvature as a functional on the set of CR-structures on a compact 3-dimensional contact manifold. Critical points of this functional can be viewed as Riemannian metrics associated to the contact structure for which the characteristic vector field generates a 1-parameter group of isometries i.e. K-contact metrics. Tanno defined a higher dimensional generalization of the Webster scalar curvature, computed the critical point condition of the corresponding integral functional and found that it is not the K-contact condition. In this paper two other generalizations are given and the critical point conditions of the corresponding integral functionals are found. For the second of these, this is the K-contact condition, suggesting that it may be the proper generalization of the Webster scalar curvature. 
Blair, D. E.; Perrone, D. A Variational Characterization of Contact Metric Manifolds With Vanishing Torsion. Canadian mathematical bulletin, Tome 35 (1992) no. 4, pp. 455-462. doi: 10.4153/CMB-1992-060-x
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     author = {Blair, D. E. and Perrone, D.},
     title = {A {Variational} {Characterization} of {Contact} {Metric} {Manifolds} {With} {Vanishing} {Torsion}},
     journal = {Canadian mathematical bulletin},
     pages = {455--462},
     year = {1992},
     volume = {35},
     number = {4},
     doi = {10.4153/CMB-1992-060-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1992-060-x/}
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