Problem In Riemannian Geometry Research Articles

On the quasisymmetric Hölder-equivalence problem for Carnot groups

A variant of Gromov's Holder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.

An example of Lichnerowicz-type Laplacian

We consider the Sampson Laplacian acting on covariant symmetric tensors on a Riemannian manifold. This operator is an example of the Lichnerowicz-type Laplacian. It is of fundamental importance in mathematical physics and appears in many problems in Riemannian geometry including the theories of infinitesimal Einstein deformations, the stability of Einstein manifolds and the Ricci flow. We study the Sampson Laplacian using the analytical method, due to Bochner, of proving vanishing theorems for the null space of a Laplace operator admitting Weitzenbock decomposition and further of estimating its lowest eigenvalue. In addition, we also survey a series of results that we obtained earlier.

The Three-body Problem in Riemannian Geometry. Hidden Irreversibility of the Classical Dynamical System

The classical three-body problem is formulated as a problem of geodesic flows on a Riemannian manifold. It is proved that a curved space allows to detect new hidden symmetries of the internal motion of a dynamical system and reduces the three-body problem to the system of 6th order. It is shown that the equivalence of the original Newtonian three-body problem and the developed representation provides coordinate transformations together with an underdetermined system of algebraic equations. The latter makes the system of geodesic equations relative to the evolution parameter (internal time), i.e. to the arc length of the geodesic curve, irreversible.

Determination of the spacetime from local time measurements

We consider an inverse problem for a Lorentzian spacetime (M, g), and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold $$\Sigma \subset M$$ determine the $$C^\infty $$ -jet of the metric in the Fermi coordinates associated to $$\Sigma $$ . We use this result to study the global determination of the spacetime (M, g) when it has a real-analytic structure or is stationary and satisfies the Einstein-scalar field equations. In addition to this, we require that (M, g) is geodesically complete modulo scalar curvature singularities. The results are Lorentzian counterparts of extensively studied inverse problems in Riemannian geometry—the determination of the jet of the metric and the boundary rigidity problem. We give also counterexamples in cases when the assumptions are not valid, and discuss inverse problems in general relativity.

Riemannian Manifold Learning

Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, i.e., how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low-dimensional space. Experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings.

Solution complète de la [formula omitted] compacité de l'ensemble des solutions de l'équation de Yamabe

On a compact Riemannian manifold ( V n , g ) ( n > 2 ) , a long-standing question is: Does the set of the solutions of the Yamabe equation compact in C 0 ? There are three cases in the Yamabe problem (see Aubin [T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, New York, 1998]) according to the sign of the inf of the Yamabe functional. We prove the C 0 compactness of the set of the solutions of the Yamabe equation (except on the sphere) in the positive case, the only case which makes difficulties.

Rational Homotopy Theory and Nonnegative Curvature

In this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non–negative curved manifolds admit (complete) metrics with non–negative curvature.

Cohomogeneity one manifolds with positive Ricci curvature

One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional -, Ricci or scalar curvature are (see [Gr]). For scalar curvature the situation is fairly well understood by comparison. Special surgery constructions as in [SY, Wr] and bundle constructions as in [Na] have resulted in a large number of interesting manifolds with positive Ricci curvature. So far the only known obstructions to have positive Ricci curvature come from obstructions to have positive scalar curvature, (see [Li] and [RS]), and from the classical Bonnet-Myers Theorem, which implies that a closed manifold with positive Ricci curvature must have finite fundamental group. It is well known that among homogeneous manifolds G/H this is also a sufficient condition (see e.g. the proof of Corollary 3.5 or [Br]). In this paper we prove that this is true as well when the manifold admits an action by a compact Lie group G with orbits of codimension one.

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≥e0, e0≡( -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.

Curvature and Symmetry of Milnor Spheres

Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He], [Po], and [Na]) and this work recently culminated in [Wr] where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive Ricci curvature. This includes all exotic spheres in dimension 7. So far, however, no example of an exotic sphere with positive sectional curvature has been found. In fact, until now, only one example of an exotic sphere with nonnegative sectional curvature was known, the so-called Gromoll-Meyer sphere [GM] in dimension 7. As one of our main results we prove:

Sub-Finslerian Metric Associated to an Optimal Control System

The problem of minimizing the cost functional of an optimal control system through the use of constrained variational calculus is a generalization of the geodetic problem in Riemannian geometry. In the framework of a geometric formulation of optimal control, we define a metric structure associated to the optimal control system on the enlarged space of state and time variables, such that the minimal length curves of the metric are the optimal solutions of the system. A twofold generalization of metric structure is applied, considering Finslerian-type metrics as well as allowed and forbidden directions (like in sub-Riemannian geometry). Free (null Hamiltonian) or fixed final parameter problems are identified with constant energy leaves, and the restriction of the metric to these leaves gives way to a family of metric structures on the usual state manifold.

Growing-Up Positive Solutions of Semilinear Elliptic Boundary Value Problems

This paper is devoted to the study of the existence, uniqueness, and asymptotic behavior of positive solutions of a class of degenerate boundary value problems for semilinear second-order elliptic differential operators which originates from the so-called Yamabe problem in Riemannian geometry. Our approach is based on the super-sub-solution method adapted to the degenerate case.

On compatibility conditions for the left Cauchy-Green deformation field in three dimensions

A fairly general sufficient condition for compatibility of the left Cauchy–Green deformation field in three dimensions has been derived. A related necessary condition is also indicated. The kinematical problem is phrased as a suitable problem in Riemannian geometry, whence the method of solution emerges naturally. The main result of the paper is general in scope and provides conditions for the existence of solutions to certain types of overdetermined systems of first-order, quasilinear partial differential equations with algebraic constraints.

Asymptotic Behavior of Radial Solutions for a Class of Semilinear Elliptic Equations

Equation (1) has its origin from, e.g., the prescribed curvature problems in Riemannian geometry, and astrophysics (i.e., the Lane Emden Fowler equation and the Matukuma equation as special cases). The asymptotic behavior of the positive solutions to (1) has recently received much attention, see e.g., [L1, LN, Na]. However, it is well-known that the above equation (1) does not always have positive solutions or positive radial solutions. In other words, under suitable conditions, radial solutions to (1) must oscillate about the zero at infinite times (see e.g., [DCC, NY, N]). Thus it becomes very interesting to know as precisely as possible the asympototic behaviors of the oscillating periods, the amplitudes of the oscillatory solutions. In this paper, we restrict our attention to the study of radial solutions to (1). More precisely, we shall discuss the asymptotic behavior of oscillatory article no. DE963208

The Yamabe Problem and Nonlinear Boundary Value Problems

We study the Yamabe problem in the context of manifolds with boundary - a basic problem in Riemannian geometry - from the point of view of nonlinear elliptic boundary value problems. By making good use of bifurcation theory from a simple eigenvalue, we show that nonpositive scalar curvatures and nonpositive mean curvatures are not always conformal to constant negative scalar curvatures and the zero mean curvature.

Equivariant geometry in EuclideanG-spheres of cohomogeneity two, I

The techniques of equivariant geometry are applied to a specific problem in Riemannian geometry, namely the study of constant mean curvature (and minimal) hypersurfaces of the Euclidean sphereS n , subject to the constraint of being invariant with respect to a suitable connected symmetry groupG ⊂ Iso(S n ), dimS n /G=2.

Black-hole uniqueness theorems in Euclidean quantum gravity

The Euclidean section of the classical Lorentzian black-hole solutions has been used in approximating the functional integral in the Euclidean path-integral approach to quantum gravity. In this paper the claim that classical black-hole uniqueness theorems apply to the Euclidean section is disproved. In particular, it is shown that although a Euclidean version of Israel's theorem does provide a type of uniqueness theorem for the Euclidean Schwarzschild solution, a Euclidean version of Robinson's theorem does not allow one to form conclusions about the uniqueness of the Euclidean Kerr solution. Despite the failure of uniqueness theorems, ''no-hair'' theorems are shown to exist. Implications are discussed. A precise mathematical statement of the Euclidean black-hole uniqueness conjecture is made and the proof, left as an unsolved problem in Riemannian geometry.

A Generalized Sphere Theorem

A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a classical theorem of Myers [10] such a manifold is compact if the sectional curvature K of M satisfies K > a > O. More precisely, the diameter d(M) of M satisfies d(M) < zc/< 8 . After the pioneering work of Rauch [11] the following result, known as the sphere theorem, was proved first by Berger [1] in even dimensions and finally by Klingenberg [8] as stated.

Conjugate Loci of Constant Order

One class of problems in riemannian geometry is to determine in what way the conjugate locus structure restricts the topological, differentiable, or metric structure of the manifold. Other than applications of the Morse theory, only a few results of this type are known. For example, if there is one point in a simply connected riemannian manifold possessing no conjugate points, then the manifold is diffeomorphic with euclidean space [12]. E. Hopf [8] has shown that a riemannian structure on the torus T2, with no conjugate pairs along any geodesic, must be flat. The only known results where one allows the existence of conjugate points are in the cases where the first conjugate locus is suitably spherical. For a 2-dimensional compact simply connected riemannian manifold M, Green [7] has shown that, if the first conjugate locus in the tangent space at each point is a sphere of the same radius, then M is isometric to the standard sphere S2. The analogous problem in higher dimensions would be for a compact simply connected riemannian manifold M, for which the first conjugate locus in each Mm is a sphere of the same radius consisting of conjugate points of maximal order. Differentiably such a manifold is a sphere, but the isometry conjecture is unsolved. In this higher dimensional case, assume the first conjugate locus is a sphere of the same radius at each point, but consisting of conjugate points of some constant order less than maximal. In this case one might conjecture the manifolds are isometric to one of the compact irreducible riemannian symmetric spaces of rank 1. However, here the topological question is still open. Allamigeon [3] has shown that, in this case, the manifold has the correct integral cohomology ring. In a somewhat different vein, Klingenberg [11] has shown that, if the conjugate locus structure in the tangent space at one point of a simply connected riemannian manifold is similar to that for one of the compact irreducible rank 1 riemannian symmetric spaces, then the manifold has the integral cohomology ring of the space after which it is modeled. In this paper, we draw some differentiable and topological conclusions from the structure of the conjugate locus at a single point. At first we do not assume the conjugate locus is spherical. If there is one point m in a compact simply connected riemannian manifold M for which each point of the

On the Index Theorem

Abstract : This paper purports to present an account of the general index problem in Riemannian geometry from a geometric point of view, to point out some of the most important index theorems, and to study some continuity questions related to the index problem. The results obtained are: 1. All self-adjoint boundary-value problems arising in Riemannian geometry can be interpreted geometrically in terms of separate endmanifolds. 2. The subsequent extension of an index theorem of Ambrose to all such problems, in particular to that associated with periodic geodesies. 3. The continuous dependence (with some exceptions) of the Ambrose conjugate points on the boundary conditions and on the Riemannian metric on the manifold.