Scalar Curvature Function Research Articles

Conformal metrics and Ricci tensors on the sphere

We consider tensors T = fg on the unit sphere S, where n > 3, g is the standard metric and f is a differentiable function on Sn. For such tensors, we consider the problems of existence of a Riemannian metric 9, conformal to g, such that Ric g = T, and the existence of such a metric that satisfies Ric g - Kg/2 = T, where K is the scalar curvature of g. We find the restrictions on the Ricci candidate for solvability, and we construct the solutions g when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors T that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor T = αg, a > 0, has no solution g on S n if α ¬= n - 1 and only metrics homothetic to g admit (n - 1)g as a Ricci tensor. We also show that if α ¬= -(n-1)(n-2)/2, then equation Ric g - Kg/2 = αg has no solution g, conformal to g on S n , and only metrics homothetic to g are solutions to this equation when a = - (n - 1)(n - 2)/2. Infinitely many C∞ solutions, globally defined on S n , are obtained for the equation -φΔ g φ + n 2|⊇ g φ| 2 -n 2(λ+φ 2 )=0, where λ ∈ R. The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on R n and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.

1/R gravity and scalar-tensor gravity

We point out that extended gravity theories, the Lagrangian of which is an arbitrary function of scalar curvature R, are equivalent to a class of the scalar-tensor theories of gravity. The corresponding gravity theory is ω=0 Brans–Dicke gravity with a potential for the Brans–Dicke scalar field, which is not compatible with solar system experiments if the field is very light: the case when such modifications become important recently.

YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS

A theorem of Escobar and Schoen asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, a necessary and sufficient condition for a C2 function K to be the scalar curvature function of some conformal metric is that K is positive somewhere. We show that for any positive C2 function K, all such metrics stay in a compact set with respect to C3 norms and the total Leray-Schauder degree of all solutions is equal to -1. Such existence and compactness results no longer hold in such generality in higher dimensions or on manifolds conformally equivalent to standard three dimensional spheres. The results are also established for more general Yamabe type equations on three dimensional manifolds.

Continuous Families of Isospectral Metrics on Simply Connected Manifolds

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S4 X S3 X S3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar curvature function changes during the deformation. Moreover, the manifolds are in general not isospectral for the Laplace operator acting on 1-forms.

Prescribing scalar curvature on Sn and related problems, part II: Existence and compactness

This is a sequel to [30], which studies the prescribing scalar curvature problem on Sn. First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and Zhang [39]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions. This counting formula, together with the compactness results, completely describes when and where blowups occur. It follows from our results that solutions to the problem may have multiple blowup points. This phenomena is new and very different from the lower-dimensional cases n = 2, 3. Next we study the problem for n ≥ 3. Some existence and compactness results have been given in [30] when the order of flatness at critical points of the prescribed scalar curvature functions K(x) is β ϵ (n − 2, n). The key point there is that for the class of K mentioned above we have completed L∞ apriori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of K(x) is β = n − 2, the L∞ estimates for solutions fail in general. In fact, two or more blowup points occur. On the other hand, we provide some existence and compactness results when the order of flatness at critical points of K(x) is β ϵ [n − 2,n). With this result, we can easily deduce that C∞ scalar curvature functions are dense in C1,α (0 < α < 1) norm among positive functions, although this is generally not true in the C2 norm. We also give a simpler proof to a Sobolev-Aubin-type inequality established in [16]. Some of the results in this paper as well as that of [30] have been announced in [29]. © 1996 John Wiley & Sons, Inc.

Conformal deformation of warped products and scalar curvature functions on open manifolds

We discuss conformal deformation and warped products on some open manifolds. We discuss how these can be applied to construct complete Riemannian metrics with specific scalar curvature functions.

Conformal deformations of prescribing scalar curvature on Riemannian manifolds with negative curvature

We study the conformal deformation for prescribing scalar curvature function\(\bar S\) on Cartan-Hadamard manifoldMn(n≥3) with strongly negative curvature. By employing the supersubsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for\(\bar S\) near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation.

A note on the Kazdan-Warner type condition

This paper addresses the necessary conditions for a function K(x) on Sn to be the scalar curvature function of a metric pointwise conformal to the standard metric on Sn. The well known necessary conditions are: K(x) is positive somewhere and satisfies the Kazdan-Warner type condition (see the text below). It has remained an outstanding problem for many years whether the above necessary conditions are also sufficient. Recently W. Chen and C. Li ([ChL]) proved that the above conditions are not sufficient by producing changing sign functions K which satisfy the above conditions, but are not scalar curvature functions of any metric pointwise conformal to the standard metric on Sn. In their construction, it is essential that K changes sign. In fact, for n = 2, it follows from the results of [XY] that for the class of positive nondegenerate axisymmetric functions the Kazdan-Warner type condition is actually necessary and sufficient. This brings up a natural question that whether this is a general fact, or it is only so for axisymmetric functions. In this note we answer the above question for 2 ≤ n ≤ 4 by producing a family of positive functions K which satisfy the Kazdan-Warner type condition, but nevertheless are not scalar curvature functions of any metric pointwise conformal to the standard metric on Sn.

A minimization problem arising from prescribing scalar curvature functions

A minimization problem arising from prescribing scalar curvature functions

Prescribing Scalar Curvature on Sn and Related Problems, Part I

Let ( S n , g 0) be the standard n-sphere. The following question was raised by L. Nirenberg. Which function K( x) on S 2 is the Gauss curvature of a metric g on S 2 conformally equivalent to g 0? Naturally one may ask a similar question in higher dimensional case, namely, which function K( x) on S n is the scalar curvature of a metric g on S n conformally equivalent to g 0? We give some apriori estimates for solutions of the prescribed scalar curvature equations for n ≥ 3 and some existence results which are quite natural extensions of previous results of Chang and Yang ([CY2]) and Bahri and Coron ([BC2]) for n = 2, 3. For n = 3 such estimates have been obtained by Schoen and Zhang (see [Sc2] and [Z]). As a byproduct of the blow up analysis completed here for n ≥ 3 we have essentially extended all the results in [Lil-3] for n = 3, 4 to higher dimensional cases. In [Lil-3], a procedure of gluing approximate solutions into genuine solutions has been completed for subcritical equations corresponding to the scalar curvature equations. This procedure is an adaptation of some "gluing technique" for periodic ode′s and periodic subcritical pde′s via variational methods originally introduced by Séré ([Se]) and Coti-Zelati and Rabinowitz ([CR1-2]). See also an earlier related paper by Coti-Zelati et al. ([CES]). Now we can pass to limit and obtain our results for n ≥ 3. In particular we show that C ∞ scalar curvature functions are C 0 dense among functions which are positive somewhere. This extends the L p density result of Bourguignon and Ezin in [BoE]. The related critical exponent equations −Δ u = K( x) u ( n+2)/( n−2) in R n with K( x) being periodic in one of the variables are also studied and infinitely many positive solutions (module translations by its periods) are obtained under some additional mild hypotheses on K( x). The main results in this paper have been announced in [Li5].

On the curvatures of Einstein spaces

For a pseudo-Riemannian manifold (M, g) of dimensionn≥3, 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.

Characterizing a class of totally real submanifolds of $S^6$ by their sectional curvatures

The first author introduced in a previous paper an important Riemannian invariant for a Riemannian manifold, namely take the scalar curvature function and subtract at each point the smallest sectional curvature at that point. He also proved a sharp inequality for this invariant for submanifolds of real space forms. In this paper we study totally real submanifolds in the nearly Kahler six-sphere that realize the equality in that inequality. In this way we characterize a class of totally real warped product immersions by one equality involving their sectional curvatures.

Prescribing Scalar Curvatures on the Conformal Classes of Complete Metrics with Negative Curvature

Let $({M^n},g)$ be a complete noncompact Riemannian manifold with the curvature bounded between two negative constants. Given a function $K$ on ${M^n}$, in terms of the behaviors of $K$ at infinite, we give a fairly complete answer to when the $K$ can be the scalar curvature function of a complete metric ${g_1}$ which is conformal to $g$.

Prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature

Let ( M n , g ) ({M^n},g) be a complete noncompact Riemannian manifold with the curvature bounded between two negative constants. Given a function K K on M n {M^n} , in terms of the behaviors of K K at infinite, we give a fairly complete answer to when the K K can be the scalar curvature function of a complete metric g 1 {g_1} which is conformal to g g .

Positive scalar curvature and periodic fundamental groups

is formed by double contraction of the riemannian curvature tensor of g (compare [He, pages 74-75]). Geometrically speaking, the scalar curvature function measures the difference between the volumes of the riemannian and euclidean geodesic disks. The existence of a riemannian metric with positive scalar curvature on a smooth manifold turns out to be of interest in many contexts. For example, by results of J. Kazdan and F. Warner [KW] the entire question of realizing a smooth function as a scalar curvature reduces to the existence of such a metric, and results of R. Schoen [Schn] on the Yamabe problem show that a metric with positive scalar curvature can be conformally deformed to one with

The metric tensor model of pattern recognition

Abstract Kolossvary, I., 1988. The metric tensor model of pattern recognition. Chemometrics and Intelligent Laboratory Systems, 4: 163–169. The classification criteria of pattern recognition methods used in analytical chemistry are based on some distance function in the N-dimensional Euclidean space. In this paper a concept based on alternative geometrical considerations is presented. It is shown that an efficient classification criterion can be defined by a new class of similarity measure, using the so-called scalar curvature function of Riemannian warped spaces. The theory and algorithm of a new pattern recognition method based on this idea, called the metric tensor model (MTM) is presented. A special version of MTM is programmed and shown to perform very well with respect to its recognition and prediction ability, on the IRIS data set.

Scalar curvature of a metric with unit volume

The problem of finding Riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of Kazdan and Warner in 1970's ([10] and references cited in it). In this paper we shall consider the same problem under a constraint on the volume. For this purpose it is useful to introduce an invariant p(M) of a smooth closed manifold M, which will be called the Yamabe number of M, defined as the supremum of #(M, C) of all conformal classes C of Riemannian metrics on M,

Scalar Curvature Functions in a Conformal Class of Metrics and Conformal Transformations

This article addresses the problem of prescribing the scalar curvature in a conformal class. (For the standard conformal class on the $2$-sphere, this is usually referred to as the Nirenberg problem.) Thanks to the action of the conformal group, integrability conditions due to J. L. Kazdan and F. W. Warner are extended, and shown to be universal. A counterexample to a conjecture by J. L. Kazdan on the role of first spherical harmonics in these integrability conditions on the standard sphere is given. Using the action of the conformal groups, some existence results are also given.

On the Scalar and Dual Formulations of the Curvature Theory of Line Trajectories

The curvature theory of ruled surfaces has been studied in two different ways. The scalar formulation proceeds by defining a seqeunce of ruled surfaces associated with the trajectory ruled surface. The relative positions of these surfaces and their distribution parameters characterize the local properties of the original ruled surface. The other formulation uses dual vector algebra to transform the differential geometry of ruled surfaces into that of spherical curves. In each theory functions are obtained which characterize the shape of the ruled surface. This paper unites these formulations by deriving formulas relating the scalar and dual curvature functions. This provides the ability to compute either set of curvature properties from either the scalar or dual vector representation of the ruled surface. The ruled surface generated by a line fixed in a body undergoing a screw displacement is examined in detail.

The coupled einstein-weyl field equations with cosmological constant and role of two higgs phenomena in weyl’s gauge model coupled to a higgs field

We study, un.der the full action of gauge invariancc including two Itiggs phenomena, Weyl’s gauge model coupled to a Higgs field. As a result, we obtain the coupled Einsteia- Weyl field equations with the cosmological constant, which are analogous to the Einstein-Maxwell equations apart from difficulties inherent to Weyl’s geometry. The vacuum solution of a Higgs field will be discussed in connection with a manifestly gaugc-invariant formulation of the Lagrangian density of gravitation. A generalized action which is a function of the gauge-invariant scalar curvature is examined.