Etude locale d'opérateurs de courbure sur l'espace hyperbolique

We show in this article that, on the unit ball in, the operators of covariant and contravariant Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the hyperbolic metric h₀. We deduce in the C^∞ case that the image of the Riemann-Christoffel curvature operator is a submanifold in a neighborhood of h₀. We deal also with some obstructions related to the asymptotic behavior of metrics near h₀, and we treat more precisely the case of the Ricci equation in dimension 2.     

Rigidity of the critical point equation

On a compact n ‐dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet. The purpose of the present paper is to prove Theorem 1, stating that a compact 3‐dimensional manifold M is isometric to the round 3‐sphere S³ if ker s^′*_g ≠ 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3‐dimensional compact manifold under certain topological conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the curvatures of Einstein spaces

For a pseudo-Riemannian manifold (M, g) of dimensionn3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofT _pM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.Partially supported by the grants from TGRC.     

On Einstein Manifolds of Positive Sectional Curvature

Let (M, g) be a compact oriented four-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M, g) is ₂, with its standard Fubini–Study metric.     

Generalized Einstein tensor for a Weyl manifold and its applications

It is well known that the Einstein tensor G for a Riemannian manifold defined by G βα = R βα 1/2 Rδβα , R βα = g βγ R γα where R γα and R are respectively the Ricci tensor and the scalar curvature of the manifold, plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work, we first obtain the generalized Einstein tensor for a Weyl manifold. Then, after studying some properties of generalized Einstein tensor, we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover, we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally, a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.     

Rigidity of noncompact complete manifolds with harmonic curvature

Let (M, g) be a noncompact complete n-manifold with harmonic curvature and positive Sobolev constant. Assume that the L ₂ norms of the traceless Ricci curvature are finite. We prove that (M, g) is Einstein if n ?? 5 and the L _n/2 norms of the Weyl curvature and traceless Ricci curvature are small enough.     

Perelman’s λ-functional and Seiberg-Witten equations

In this paper, we estimate the supremum of Perelman’s λ-functional λ _M (g) on Riemannian 4-manifold (M, g) by using the Seiberg-Witten equations. Among other things, we prove that, for a compact Kähler-Einstein complex surface (M, J, g ₀) with negative scalar curvature, (i) if g ₁ is a Riemannian metric on M with λ _M (g ₁) = λ _M (g ₀), then $Vol_{g_1 } $ (M) ? $Vol_{g_0 } $ (M). Moreover, the equality holds if and only if g ₁ is also a Kähler-Einstein metric with negative scalar curvature. (ii) If {g _t}, t ∈ [?1, 1], is a family of Einstein metrics on M with initial metric g ₀, then g _t is a Kähler-Einstein metric with negative scalar curvature.     

Prescription de la courbure de Ricci au voisinage de la métrique hyperbolique

On the unit ball of, one considers the standard hyperbolic metric H₀ whose Ricci curvature equals R₀ and Riemann-Christoffel curvature is. We prove that, for any symmetric tensor R near R₀, there exists a unique metric H near H₀ whose Ricci curvature is R. We deduce in the C^∞ case that the image of the Riemann-Christoffel operator is a submanifold in a neighborhood of. Finally, we study more precisely the Ricci equation in dimension 2.     

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Let (M, g) be a 4-dimensional Einstein Riemannian manifold. At each point p of M, the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor R at p. In this basis, up to standard symmetries and antisymmetries, just 5 components of the curvature tensor R are nonzero. For the space of constant curvature, the group O(4) acts as a transformation group between ST bases at T _p M and for the so-called 2-stein curvature tensors, the group Sp(1) ? SO(4) acts as a transformation group between ST bases. In the present work, the complete list of Lie subgroups of SO(4) which act as transformation groups between ST bases for certain classes of Einstein curvature tensors is presented. Special representations of groups SO(2), T ², Sp(1) or U(2) are obtained and the classes of curvature tensors whose transformation group into new ST bases is one of the mentioned groups are determined.     

Einstein Metrics with Nonpositive Sectional Curvature on Extensions of Lie Algebras of Heisenberg Type

The purpose of this article is to study some simply connected Lie groups with left invariant Einstein metric, negative Einstein constant and nonpositive sectional curvature. These Lie groups are classified if their associated metric Lie algebra s is of Iwasawa type and s = An₁n₂...n_r, where all n_iare Lie algebras of Heisenberg type with [[n_i,n_j] = {0} for ij. The most important ideas of the article are based on a construction method for Einstein spaces introduced by Wolter in 1991. By this method some new examples of Einstein spaces with nonpositive curvature are constructed. In another part of the article it is shown that Damek-Ricci spaces have negative sectional curvature if and only if they are symmetric spaces.     

Infinitesimal harmonic transformations and ricci solitons on complete Riemannian manifolds

Ricci solitons were introduced by R. Hamilton as natural generalizations of Einstein metrics. A Ricci soliton on a smooth manifold M is a triple (g₀,ξ, λ), where g₀ is a complete Riemannian metric, ξ a vector field, and λ a constant such that the Ricci tensor Ric₀ of the metric g₀ satisfies the equation ò2 Ric₀ = L_ξg₀ + 2λ_go. The following statement is one of the main results of the paper. Let (g₀,ξ, λ) be a Ricci soliton such that M,g₀ is a complete noncompact oriented Riemannian manifold, $ \int\limits_M {\left\| \xi \right\|dv < \infty } $ \int\limits_M {\left\| \xi \right\|dv < \infty } , and the scalar curvature s₀ of g₀ has a constant sign on M, then (M, g₀) is an Einstein manifold     

The nonvacuum Einstein flow on surfaces of nonnegative curvature

We prove future nonlinear stability of homogeneous solutions to the Einstein–Vlasov system with massive particles on manifolds with topology M = ?×Σ, where Σ is either 𝕊² or 𝕋². For the sphere this implies the existence of an open subset of the initial data manifold with elements of strictly positive scalar curvature, whose developments are future geodesically complete. In combination with an earlier result for hyperbolic surfaces we conclude future completeness for the Einstein–Vlasov system in 2+1 dimensions independent of the compact spatial topology for an open set of initial data.     

Homotopy finiteness theorems and Helly’s theorem

A new proof is presented of the noncriticality of the distance function on a neighborhood of the diagonal inM ×M, whereM is a compactn-dimensional manifold with diameter ≤D ₀, volume ≥v ₀ > 0, and sectional curvature ≥ ?Λ, with the size of the neighborhood depending only onn,D ₀,v ₀, and Λ. This gives a generalization of the Grove-Petersen finiteness theorem and elucidates the role of the sectional curvature bound, as opposed to just a Ricci curvature lower bound, in such theorems.     

Einstein Finsler metrics and killing vector fields on Riemannian manifolds

We use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on S ³ with Ric = 2F ², Ric = 0 and Ric = -2F ², respectively. This family of metrics provides an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.     

Rigidity of Einstein 4-manifolds with positive curvature

An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥ε₀, ε₀≡(-23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S ⁴, RP ⁴ with constant sectional curvature K=1/3, or CP ² with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions on S ⁴ and CP ² contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S ⁴, RP ⁴, or CP ² but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions. Oblatum 4-II-1999 & 4-V-2000?Published online: 16 August 2000     

Geodesics in minimal surfaces

Abstract—In this paper, we consider connected minimal surfaces in R³ with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature k _g ¹ and k _g ² of the coordinates curves satisfy αk _g ¹ + βk _g ² = 0, α, β ∈ R.     

Seiberg–Witten Invariants of Nonsimple Type and Einstein Metrics

We study the behavior of the moduli space of solutions to theSeiberg–Witten equations under a conformal change in the metric of aKähler surface (M,g). If the canonical line bundle K _M is ofpositive degree, we prove there is only one (up to gauge) solution tothe equations associated to any conformal metric to g. We use this, toconstruct examples of four dimensional manifolds withSpin ^c -structures, whose moduli spaces of solutions to theSeiberg–Witten equations, represent a nontrivial bordism class ofpositive dimension, i.e. the Spin ^c -structures are not inducedby almost complex structures. As an application, we show the existenceof infinitely many nonhomeomorphic compact oriented 4-manifolds withfree fundamental group and predetermined Euler characteristic andsignature that do not carry Einstein metrics.     

Einstein structures: Existence versus uniqueness

Two well-known questions in differential geometry are “Does every compact manifold of dimension greater than four admit an Einstein metric?” and “Does an Einstein metric of a negative scalar curvature exist on a sphere?” We demonstrate that these questions are related: For everyn≥5 the existence of metrics for which the deviation from being Einstein is arbitrarily small on every compact manifold of dimensionn (or even on every smooth homology sphere of dimensionn) implies the existence of metrics of negative Ricci curvature on the sphereS ⁿ for which the deviation from being Einstein is arbitrarily small. Furthermore, assuming either a version of the Palais-Smale condition or the plausible looking existence of an algorithm deciding when a given metric on a compact manifold is close to an Einstein metric, we show for anyn≥5 that: 1) If everyn-dimensional smooth homology sphere admits an Einstein metric thenS ⁿ admits infinitely many Einstein structures of volume one and of negative scalar curvature; 2) If every compactn-dimensional manifold admits an Einstein metric then every compactn-dimensional manifold admits infinitely many distinct Einstein structures of volume one and of negative scalar curvature.     

On transversally symmetric pseudo-Einstein and Fefferman–Einstein spaces

We describe and construct pseudo-Hermitian structures θ without torsion (i.e. with transverse symmetry) whose Webster–Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures TS-pseudo-Einstein and our first result states that all these structures can locally be derived from Kähler–Einstein metrics. Then we discuss the corresponding Fefferman metrics of the TS-pseudo-Einstein structures. These are never Einstein. However, our second result states that they are locally always conformally Einstein.