Geodesic
The Einstein–Hilbert type action on pseudo-Riemannian almost-product manifolds
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019) no. 1, pp. 86-121 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We develop variation formulas for the quantities of extrinsic geometry of almost-product pseudo-Riemannian manifolds, and we consider the variations of metric preserving orthogonality of the distributions. These formulas are applied to study the Einstein–Hilbert type actions for the mixed scalar curvature and the extrinsic scalar curvature of a distribution. The Euler–Lagrange equations for these variations are derived in full generality and in several particular cases (foliations that are integrable plane fields, conformal submersions, etc.). The obtained Euler–Lagrange equations generalize the results for codimension-one foliations to the case of arbitrary codimension, and admit a number of solutions, e.g., twisted products and isoparametric foliations. 
@article{JMAG_2019_15_1_a3,
     author = {Vladimir Rovenskiǐ and Tomasz Zawadzki},
     title = {The {Einstein{\textendash}Hilbert} type action on {pseudo-Riemannian} almost-product manifolds},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {86--121},
     year = {2019},
     volume = {15},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2019_15_1_a3/}
}
TY  - JOUR
AU  - Vladimir Rovenskiǐ
AU  - Tomasz Zawadzki
TI  - The Einstein–Hilbert type action on pseudo-Riemannian almost-product manifolds
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2019
SP  - 86
EP  - 121
VL  - 15
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JMAG_2019_15_1_a3/
LA  - en
ID  - JMAG_2019_15_1_a3
ER  - 
%0 Journal Article
%A Vladimir Rovenskiǐ
%A Tomasz Zawadzki
%T The Einstein–Hilbert type action on pseudo-Riemannian almost-product manifolds
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2019
%P 86-121
%V 15
%N 1
%U http://geodesic.mathdoc.fr/item/JMAG_2019_15_1_a3/
%G en
%F JMAG_2019_15_1_a3
Vladimir Rovenskiǐ; Tomasz Zawadzki. The Einstein–Hilbert type action on pseudo-Riemannian almost-product manifolds. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019) no. 1, pp. 86-121. http://geodesic.mathdoc.fr/item/JMAG_2019_15_1_a3/

[1] E. Barletta, S. Dragomir, V. Rovenski, M. Soret, “Mixed gravitational field equations on globally hyperbolic spacetimes”, Class. Quantum Grav., 30 (2013), 085015 | DOI | MR | Zbl

[2] E. Barletta, S. Dragomir, V. Rovenski, “The Einstein–Hilbert type action on foliations”, Balkan J. Geom. Appl., 22:1 (2017), 1–17 | MR | Zbl

[3] A. Bejancu, H. Farran, Foliations and Geometric Structures, Mathematics and Its Applications, 580, Springer-Verlag, Dordrecht, 2006 | MR | Zbl

[4] A. Candel, L. Conlon, Foliations, v. I, Graduate Studies in Mathematics, 23, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl

[5] M. Falcitelli, S. Ianus, A.M. Pastore, Riemannian Submersions and Related Topics, World Scientific, Singapore, 2004 | MR | Zbl

[6] H. Gluck, W. Ziller, “On the volume of a unit vector field on the three-sphere”, Comment. Math. Helv., 61 (1986), 177–192 | DOI | MR | Zbl

[7] A. Gray, “Pseudo-Riemannian almost-product manifolds and submersions”, J. Math. Mech., 16:7 (1967), 715–737 | MR | Zbl

[8] S. Gudmundsson, “On the geometry of harmonic morphisms”, Math. Proc. Cambridge Philos. Soc., 108 (1990), 461–466 | DOI | MR | Zbl

[9] P. Li and L.-F. Tam, “Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set”, Ann. of Math. (2), 125:1 (1987), 171–207 | DOI | MR | Zbl

[10] A.M. Naveira, “A classification of Riemannian almost-product manifolds”, Rend. Mat., 7:3 (1983), 577–592 | MR

[11] B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, New York, 1983 | MR | Zbl

[12] R. Ponge, H. Reckziegel, “Twisted products in pseudo-Riemannian geometry”, Geom. Dedicata, 48 (1993), 15–25 | DOI | MR | Zbl

[13] V. Rovenski, “On solutions to equations with partial Ricci curvature”, J. Geom. Phys., 86 (2014), 370–382 | DOI | MR | Zbl

[14] V. Rovenski, P. Walczak, Topics in Extrinsic Geometry of Codimension-One Foliations, Springer Briefs in Mathematics, Springer, New York, 2011 | DOI | MR | Zbl

[15] P. Tondeur, Foliations on Riemannian Manifolds, Springer-Verlag, New York, 1988 | MR | Zbl

[16] P. Topping, Lectures on the Ricci Flow, LMS Lecture Notes, 325, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl

[17] P. Walczak, “An integral formula for a Riemannian manifold with two orthogonal complementary distributions”, Colloq. Math., 58 (1990), 243–252 | DOI | MR | Zbl

[18] T. Zawadzki, “Existence conditions for conformal submersions with totally umbilical fibers”, Differential Geom. Appl., 35 (2014), 69–85 | DOI | MR | Zbl