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11
Renormalizing curvature integrals on PoincaréEinstein manifolds
, 2005
"... After analyzing renormalization schemes on a PoincaréEinstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is wellknown, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms ..."
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Cited by 27 (5 self)
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After analyzing renormalization schemes on a PoincaréEinstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is wellknown, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the PoincaréEinstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the GaussBonnet theorem.
On (2k)Minimal Submanifolds
, 706
"... Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)th GaussBonnet curvature function, called ..."
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Cited by 5 (3 self)
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Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)th GaussBonnet curvature function, called (2k)minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the (2k + 1)GaussBonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to (2k)minimal submanifolds.
On Generalized Einstein Metrics
, 2008
"... Recall that the usual Einstein metrics are those for which the first Ricci contraction of the covariant Riemann curvature tensor is proportional to the metric. Assuming the same type of restrictions but instead on the different contractions of Thorpe tensors, one gets several natural generalizations ..."
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Cited by 4 (2 self)
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Recall that the usual Einstein metrics are those for which the first Ricci contraction of the covariant Riemann curvature tensor is proportional to the metric. Assuming the same type of restrictions but instead on the different contractions of Thorpe tensors, one gets several natural generalizations of Einstein’s condition. In this paper, we study some properties of these classes of metrics.
Remarks on generalized Einstein manifolds
"... We study some properties of generalized Einstein manifolds. In particular, we prove the nonnegativity of the EulerPoincaré characterestic for (4k)dimensional compact hyper (2k)Einstein manifolds generalizing a well known result of Berger about four dimensional usual Einstein manifolds. ..."
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Cited by 3 (3 self)
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We study some properties of generalized Einstein manifolds. In particular, we prove the nonnegativity of the EulerPoincaré characterestic for (4k)dimensional compact hyper (2k)Einstein manifolds generalizing a well known result of Berger about four dimensional usual Einstein manifolds.
On Gauss–Bonnet Curvatures
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2007
"... The (2k)th Gauss–Bonnet curvature is a generalization to higher dimensions of the (2k)dimensional Gauss–Bonnet integrand, it coincides with the usual scalar curvature for k = 1. The Gauss–Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where ..."
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Cited by 1 (1 self)
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The (2k)th Gauss–Bonnet curvature is a generalization to higher dimensions of the (2k)dimensional Gauss–Bonnet integrand, it coincides with the usual scalar curvature for k = 1. The Gauss–Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss–Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metrics and minimal submanifolds. Key words: Gauss–Bonnet curvatures; Gauss–Bonnet gravity; lovelock gravity; generalized Einstein metrics; generalized minimal submanifolds; generalized Yamabe problem 2000 Mathematics Subject Classification: 53C20; 53C25 1 An introduction to Gauss–Bonnet curvatures We shall present in this section several approaches to the Gauss–Bonnet curvatures. For precise definitions and examples the reader is encouraged to consult [17, 18, 20]. 1.1 Gauss–Bonnet curvatures vs. curvature invariants of Weyl’s tube formula In a celebrated paper [38] published in 1939, Hermann Weyl proved that the volume of a tube of radius r around an embedded compact psubmanifold M of the ndimensional Euclidean space is a polynomial in the radius of the tube as follows: Vol(tube(r)) = [p/2] i=0
On Weitzenböck curvature operators
"... The Weitzenböck curvature operators are the curvature terms of order zero that appear in the well known classical Weitzenböck formula. In this paper, we use the formalism of double forms to prove a simple formula for this operators and to study their geometric properties. ..."
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The Weitzenböck curvature operators are the curvature terms of order zero that appear in the well known classical Weitzenböck formula. In this paper, we use the formalism of double forms to prove a simple formula for this operators and to study their geometric properties.
On Weitzenbök Curvature Operators
, 2006
"... The Weitzenbök curvature operators are the curvature terms of order zero that appear in the well known classical Weitzenbök formula. In this paper, we use the formalism of double forms to prove a simple formula for this operators and to study their geometric properties. ..."
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The Weitzenbök curvature operators are the curvature terms of order zero that appear in the well known classical Weitzenbök formula. In this paper, we use the formalism of double forms to prove a simple formula for this operators and to study their geometric properties.
Variational Properties of the GaussBonnet
, 2007
"... The GaussBonnet curvature of order 2k is a generalization to higher dimensions of the GaussBonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional GaussBonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as f ..."
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The GaussBonnet curvature of order 2k is a generalization to higher dimensions of the GaussBonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional GaussBonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant GaussBonnet curvature.