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1
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Integral formulae for a Riemannian manifold with two orthogonal distributions

100%
Rovenski V.
Open Mathematics
|
2011
|
tom 9
|
nr 3
558-577
EN
We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.
2
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Total curvature and volume of foliations on the sphere S 2

81%
Fawaz A.
Open Mathematics
|
2009
|
tom 7
|
nr 4
660-669
EN
In this paper we study a curvature integral associated with a pair of orthogonal foliations on the Riemann sphere S 2 and we compute the minimal value of the volume of meromorphic foliations.
3
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Harmonic conformal flows on manifolds of constant curvature

81%
Fawaz A.
Open Mathematics
|
2007
|
tom 5
|
nr 3
493-504
EN
We compute the energy of conformal flows on Riemannian manifolds and we prove that conformal flows on manifolds of constant curvature are critical if and only if they are isometric.
4
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Deforming metrics of foliations

81%
Rovenski V., Wolak R.
Open Mathematics
|
2013
|
tom 11
|
nr 6
1039-1055
EN
Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.
5
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The automorphism groups of foliations with transverse linear connection

62%
Zhukova N., Dolgonosova A.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2076-2088
EN
The category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.
6
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A spectral estimate for the Dirac operator on Riemannian flows

52%
Ginoux N., Habib G.
Open Mathematics
|
2010
|
tom 8
|
nr 5
950-965
EN
We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.
7
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On localization in holomorphic equivariant cohomology

42%
Bruzzo U., Rubtsov V.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1442-1454
EN
We study a holomorphic equivariant cohomology built out of the Atiyah algebroid of an equivariant holomorphic vector bundle and prove a related localization formula. This encompasses various residue formulas in complex geometry, in particular we shall show that it contains as special cases Carrell-Liebermann’s and Feng-Ma’s residue formulas, and Baum-Bott’s formula for the zeroes of a meromorphic vector field.
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