Showing papers on "Riemann curvature tensor published in 2007"
TL;DR: In this article, a measure contraction property of metric measure spaces is introduced, which can be regarded as a generalized notion of the lower Ricci curvature bound on Riemannian manifolds.
Abstract: We introduce a measure contraction property of metric measure spaces which can be regarded as a generalized notion of the lower Ricci curvature bound on Riemannian manifolds. It is actually equivalent to the lower bound of the Ricci curvature in the Riemannian case. We will generalize the Bonnet?Myers theorem, and prove that this property is preserved under the measured Gromov?Hausdorff convergence.
284 citations
TL;DR: In this paper, an entropy functional for any vector field in terms of a fourth-rank divergence-free tensor with the symmetries of the curvature tensor was introduced.
Abstract: The null surfaces of a spacetime act as oneway membranes and can block information for a corresponding family of observers (timelike curves). Since lack of information can be related to entropy, this suggests the possibility of assigning an entropy to the null surfaces of a spacetime. We motivate and introduce such an entropy functional for any vector field in terms of a fourth-rank divergence-free tensor ${P}_{ab}^{cd}$ with the symmetries of the curvature tensor. Extremizing this entropy for all the null surfaces then leads to equations for the background metric of the spacetime. When ${P}_{ab}^{cd}$ is constructed from the metric alone, these equations are identical to Einstein's equations with an undetermined cosmological constant (which arises as an integration constant). More generally, if ${P}_{ab}^{cd}$ is allowed to depend on both metric and curvature in a polynomial form, one recovers the Lanczos-Lovelock gravity. In all these cases: (a) We only need to extremize the entropy associated with the null surfaces; the metric is not a dynamical variable in this approach. (b) The extremal value of the entropy agrees with standard results, when evaluated on shell for a solution admitting a horizon. The role of the full quantum theory of gravity will be to provide the specific form of ${P}_{ab}^{cd}$ which should be used in the entropy functional. With such an interpretation, it seems reasonable to interpret the Lanczos-Lovelock type terms as quantum corrections to classical gravity.
173 citations
TL;DR: This article presents an interactive design system that allows a user to create a wide variety of symmetric tensor fields over 3D surfaces either from scratch or by modifying a meaningful input tensor field such as the curvature tensor.
Abstract: Designing tensor fields in the plane and on surfaces is a necessary task in many graphics applications, such as painterly rendering, pen-and-ink sketching of smooth surfaces, and anisotropic remeshing. In this article, we present an interactive design system that allows a user to create a wide variety of symmetric tensor fields over 3D surfaces either from scratch or by modifying a meaningful input tensor field such as the curvature tensor. Our system converts each user specification into a basis tensor field and combines them with the input field to make an initial tensor field. However, such a field often contains unwanted degenerate points which cannot always be eliminated due to topological constraints of the underlying surface. To reduce the artifacts caused by these degenerate points, our system allows the user to move a degenerate point or to cancel a pair of degenerate points that have opposite tensor indices. These operations provide control over the number and location of the degenerate points in the field. We observe that a tensor field can be locally converted into a vector field so that there is a one-to-one correspondence between the set of degenerate points in the tensor field and the set of singularities in the vector field. This conversion allows us to effectively perform degenerate point pair cancellation and movement by using similar operations for vector fields. In addition, we adapt the image-based flow visualization technique to tensor fields, therefore allowing interactive display of tensor fields on surfaces. We demonstrate the capabilities of our tensor field design system with painterly rendering, pen-and-ink sketching of surfaces, and anisotropic remeshing
136 citations
TL;DR: In this paper, a higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n.
Abstract: A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension n. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in n − 4 spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers–Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.
130 citations
TL;DR: In this article, the authors studied the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor and proved an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature.
Abstract: We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
121 citations
TL;DR: In this article, it was shown that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, Ii, D or O, provided additional conditions are fulfilled, as for most known black hole/ring solutions.
Abstract: We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, Ii, D or O. This also applies to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons, on which we briefly comment.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two-dimensional Lorentzian factor are restricted to the types D or O. By exploring algebraic consequences of the Bianchi identities, we then analyze the simplest non-trivial case from the above classes?type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for 'generic' type D vacuum spacetimes (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact also applies to type II spacetimes). For n ? 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n ? 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 ? 3 real matrix ?ij. In the case with 'twistfree' (Aij = 0) principal null geodesics we show that in a 'generic' case ?ij is symmetric and eigenvectors of ?ij coincide with eigenvectors of the expansion matrix Sij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that ?ij is symmetric. The five-dimensional Myers?Perry black hole and Kerr?NUT?AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes.
116 citations
TL;DR: In this article, the intrinsic torsion of a SU(3)-manifold was studied and a number of formulae for the Ricci and the scalar curvature in terms of torsions forms were derived.
110 citations
TL;DR: The Invar package is introduced, a fast manipulator of generic scalar polynomial expressions formed from the Riemann tensor of a four-dimensional metric-compatible connection, which has been implemented both in Mathematica and Maple algebraic systems.
104 citations
Posted Content•
TL;DR: For a survey of known manifolds with non-negative sectional curvature, see as discussed by the authors and the survey by Burkhard Wilking in this volume, where the authors focus on the description of known examples and provide suggestions where to look for new ones.
Abstract: Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on S × S, also fit into this subject. For non-negatively curved manifolds, there are a number of obstruction theorems known, see Section 1 below and the survey by Burkhard Wilking in this volume. It is somewhat surprising that the only further obstructions to positive curvature are given by the classical Bonnet-Myers and Synge theorems on the fundamental group. Although there are many examples with non-negative curvature, they all come from two basic constructions, apart from taking products. One is taking an isometric quotient of a compact Lie group equipped with a biinvariant metric and another a gluing procedure due to Cheeger and recently significantly generalized by Grove-Ziller. The latter examples include a rich class of manifolds, and give rise to non-negative curvature on many exotic 7-spheres. On the other hand, known manifolds with positive sectional curvature are very rare, and are all given by quotients of compact Lie groups, and, apart from the classical rank one symmetric spaces, only exist in dimension below 25. Due to this lack of knowledge, it is therefore of importance to discuss and understand known examples and find new ones. In this survey we will concentrate on the description of known examples, although the last section also contains suggestions where to look for new ones. The techniques used to construct them are fairly simple. In addition to the above, the main tool is a deformation described by Cheeger that, when applied to nonnegatively curved manifolds, tends to increase curvature. Such Cheeger deformations can be considered as the unifying theme of this survey. We can thus be fairly explicit in the proof of the existence of all known examples which should make the basic material understandable at an advanced graduate student level. It is the hope of this author that it will thus encourage others to study this beautiful subject. This survey originated in the Rudolph Lipschitz lecture series the author gave at the University of Bonn in 2001 and various courses taught at the University of Pennsylvania.
104 citations
30 Aug 2007
TL;DR: In this article, it was shown that a complete Riemannian metric of nonnegative sectional curvature can be deformed to a metric of positive Ricci curvature.
Abstract: In this paper we address the question whether a complete Riemannian metric of nonnegative sectional curvature can be deformed to a metric of positive Ricci curvature. This problem came up implicitly in various recent new constructions for metrics with positive Ricci curvature. Grove and Ziller [GZ] showed that any compact cohomogeneity one manifold with finite fundamental group admits invariant metrics with positive Ricci curvature. The case that both non-regular orbits have codimension two is especially resilient. By earlier work of Grove and Ziller it has been known that these manifolds admit invariant nonnegatively curved metrics. However, in certain cases the Ricci curvature of these metrics is not positive at any point and hence they cannot apply the deformation theorem of Aubin [A] and Ehrlich [E]: a metric of nonnegative Ricci curvature is conformally equivalent to a metric with positive Ricci curvature if and only if the Ricci curvature is positive at some point. Similar problems arise in the work of Schwachhöfer and Tuschmann on quotient spaces [ST]. Our main result is: Theorem A. Let (Mn, g) be a compact Riemannian manifold with finite fundamental group and nonnegative sectional curvature. Then Mn admits a metric with positive Ricci curvature.
100 citations
TL;DR: In this article, the Ricci tensor of a real hypersurface M in complex two-plane Grassmannians G 2 (C m +2 ) was derived from the equation of Gauss.
Abstract: . In this paper, flrst we introduce the full expression of thecurvature tensor of a real hypersurface M in complex two-plane Grass-mannians G 2 (C m +2 ) from the equation of Gauss and derive a new formulafor the Ricci tensor of M in G 2 (C m +2 ). Next we prove that there do notexist any Hopf real hypersurfaces in complex two-plane Grassmannians G 2 (C m +2 ) with parallel and commuting Ricci tensor. Finally we showthat there do not exist any Einstein Hopf hypersurfaces in G 2 (C m +2 ). IntroductionIn the geometry of real hypersurfaces in complex space forms or in quater-nionic space forms it can be easily checked that there do not exist any realhypersurfaces with parallel shape operator A by virtue of the equation of Co-dazzi.But if we consider a real hypersurface with parallel Ricci tensor S in suchspace forms, the proof of its non-existence is not so easy. In the class of Hopfhypersurfaces Kimura [7] has asserted that there do not exist any real hyper-surfaces in a complex projective space C
TL;DR: Using a new type of Jacobi field estimate, this article proved a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.
Abstract: Using a new type of Jacobi field estimate we will prove a duality theorem for singular Riemannian foliations in complete manifolds of nonnegative sectional curvature.
TL;DR: In this article, the Euler class and the Pontryagin class are used to characterize the topological properties of Riemannian space-time in the presence of spin.
Abstract: Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.
TL;DR: In this paper, the authors generalize Shi and Tam's inequality to the case when the Gaussian curvature of the surface is allowed to be negative and obtain a future-directed time-like quasi-local energy-momentum.
Abstract: In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannian version corresponds to an earlier theorem of Shi and Tam \cite{st}. In this article, we generalize such an inequality to the case when the Gaussian curvature of the surface is allowed to be negative. This is done by an isometric embedding into the hyperboloid in the Minkowski space and a future-directed time-like quasi-local energy-momentum is obtained.
18 Oct 2007
TL;DR: In this article, it was shown that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite.
Abstract: We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case.
TL;DR: A framework for decomposing variations in diffusion tensors into changes in shape and orientation is introduced, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.
Abstract: Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.
TL;DR: In this article, a higher order homogenization scheme based on non-linear micropolar kinematics representing the macroscopic variation within a representative volume element (RVE) of the material is presented.
Abstract: The paper presents a higher order homogenization scheme based on non-linear micropolar kinematics representing the macroscopic variation within a representative volume element (RVE) of the material. On the microstructural level the micro–macro kinematical coupling is introduced as a second-order Taylor series expansion of the macro displacement field, and the microstructural displacement variation is gathered in a fluctuation term. This approach relates strongly to second gradient continuum formulations, presented by, e.g. Kouznetsova et al. (Int. J. Numer. Meth. Engng 2002; 54:1235–1260), thus establishing a link between second gradient and micropolar theories. The major difference of the present approach as compared to second gradient formulations is that an additional constraint is placed on the higher order deformation gradient in terms of the micropolar stretch. The driving vehicle for the derivation of the homogenized macroscopic stress measures is the Hill–Mandel condition, postulating the equivalence of microscopic and macroscopic (homogenized) virtual work. Thereby, the resulting homogenization procedure yields not only a stress tensor, conjugated to the micropolar stretch tensor, but also the couple stress tensor, conjugated to the micropolar curvature tensor. The paper is concluded by a couple of numerical examples demonstrating the size effects imposed by the homogenization of stresses based on the micropolar kinematics. Copyright © 2006 John Wiley & Sons, Ltd.
29 Oct 2007
TL;DR: A new class of interpolation paths for tensors is proposed, termed geodesic-loxodromes, which explicitly preserve clinically important tensor attributes, such as mean diffusivity or fractional anisotropy, while using basic differential geometry to interpolate tensor orientation.
Abstract: In algorithms for processing diffusion tensor images, two common ingredients are interpolating tensors, and measuring the distance between them. We propose a new class of interpolation paths for tensors, termed geodesic-loxodromes, which explicitly preserve clinically important tensor attributes, such as mean diffusivity or fractional anisotropy, while using basic differential geometry to interpolate tensor orientation. This contrasts with previous Riemannian and Log-Euclidean methods that preserve the determinant. Path integrals of tangents of geodesic-loxodromes generate novel measures of over-all difference between two tensors, and of difference in shape and in orientation.
TL;DR: In this paper, the authors consider Lorentzian manifolds with distributional curvature tensors and derive the jump relations associated with singular parts of connection and curvature operators.
Abstract: Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singu- lar patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate- free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the frame- work of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we inves- tigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curva- ture tensors and the second fundamental form of the hypersurface and to establish the Gauss-Codazzi equations.
TL;DR: In this paper, the authors show boundedness of the fundamental group under a global L p control (for p > n / 2 ) of the Ricci curvature, and show that complete metrics with similar L n 2 -control of their curvatures are dense in the set of complete metrics of any compact differentiable manifold.
Abstract: We show boundedness of the diameter and finiteness of the fundamental group under a global L p control (for p > n / 2 ) of the Ricci curvature. Conversely, metrics with similar L n 2 -control of their Ricci curvature are dense in the set of complete metrics of any compact differentiable manifold.
TL;DR: In this paper, the analysis of heterotic string compactifications on half-flat mirror manifolds was extended to include the 10-dimensional gauge fields, and the results were also extended to more general manifolds with SU(3) structure.
Abstract: In this paper, we continue the analysis of heterotic string compactifications on half-flat mirror manifolds by including the 10-dimensional gauge fields. It is argued, that the heterotic Bianchi identity is solved by a variant of the standard embedding. Then, the resulting gauge group in four dimensions is still E6 despite the fact that the Levi-Civita connection has SO(6) holonomy. We derive the associated four-dimensional effective theories including matter field terms for such compactifications. The results are also extended to more general manifolds with SU(3) structure.
TL;DR: In this article, the conditions under which a generic supergravity model involving chiral and vector multiplets can admit viable metastable vacua with spontaneously broken supersymmetry and realistic cosmological constant were studied.
Abstract: We study the conditions under which a generic supergravity model involving chiral and vector multiplets can admit viable metastable vacua with spontaneously broken supersymmetry and realistic cosmological constant. To do so, we impose that on the vacuum the scalar potential and all its first derivatives vanish, and derive a necessary condition for the matrix of its second derivatives to be positive definite. We study then the constraints set by the combination of the flatness condition needed for the tuning of the cosmological constant and the stability condition that is necessary to avoid unstable modes. We find that the existence of such a viable vacuum implies a condition involving the curvature tensor for the scalar geometry and the charge and mass matrices for the vector fields. Moreover, for given curvature, charges and masses satisfying this constraint, the vector of F and D auxiliary fields defining the Goldstino direction is constrained to lie within a certain domain. The effect of vector multiplets relative to chiral multiplets is maximal when the masses of the vector fields are comparable to the gravitino mass. When the masses are instead much larger or much smaller than the gravitino mass, the effect becomes small and translates into a correction to the effective curvature. We finally apply our results to some simple classes of examples, to illustrate their relevance.
TL;DR: In this article, the authors discuss linear connections and curvature tensors in the context of geometry of parallelizable manifolds (or absolute parallelism geometry) using Bianchi identities.
TL;DR: In this article, the geometry of B-manifolds is studied, and some properties of Riemannian curvature tensors of paraholomorphic B-mansifolds are given.
TL;DR: In this article, the authors constructed 2-surfaces of prescribed mean curvature in 3-manifolds carrying asymptotically flat initial data for an isolated gravitating system with rather general decay conditions.
Abstract: We construct 2-surfaces of prescribed mean curvature in 3manifolds carrying asymptotically flat initial data for an isolated gravitating system with rather general decay conditions. The surfaces in question form a regular foliation of the asymptotic region of such a manifold. We recover physically relevant data, especially the ADM-momentum, from the geometry of the foliation. For a given set of data (M, g, K), with a three dimensional manifold M, its Riemannian metric g, and the second fundamental form K in the surrounding four dimensional Lorentz space time manifold, the equation we solve is H+P = const or H −P = const. Here H is the mean curvature, and P = trK is the 2-trace of K along the solution surface. This is a degenerate elliptic equation for the position of the surface. It prescribes the mean curvature anisotropically, since P depends on the direction of the normal.
04 Sep 2007
TL;DR: A theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces--discrete surface Ricci flow, whose continuous counter part has been used in the proof of Poincare conjecture.
Abstract: Conformal geometry is at the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields.
This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces--discrete surface Ricci flow, whose continuous counter part has been used in the proof of Poincare conjecture. Continuous Ricci flow conformally deforms a Riemannian metric on a smooth surface such that the Gaussian curvature evolves like a heat diffusion process. Eventually, the Gaussian curvature becomes constant and the limiting Riemannian metric is conformal to the original one.
In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. The existence and uniqueness of the solution and the convergence of the flow process are theoretically proven, and numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow are also designed.
Discrete Ricci flow has broad applications in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces.
TL;DR: In this article, the uniqueness of mean curvature flow of general codimensions and general ambient manifolds has been studied and a pseudolocality theorem for the Ricci flow has been proved.
Abstract: Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.
Book•
26 Apr 2007
TL;DR: The Geometry of the Riemann Curvature TensorCurvature Homogeneous Generalized Plane Wave Manifolds Other Pseudo-Riemannian Manifold The Curvatures Tensor Complex Osserman AlgebraicCurvatures Tensors Stanilov-Tsankov Theory as mentioned in this paper.
Abstract: The Geometry of the Riemann Curvature Tensor Curvature Homogeneous Generalized Plane Wave Manifolds Other Pseudo-Riemannian Manifolds The Curvature Tensor Complex Osserman Algebraic Curvature Tensors Stanilov-Tsankov Theory.
TL;DR: This paper focuses on the contribution of algorithms and tools developed to deal with tensors to the anisotropic filtering and regularization of tensor fields and presents promising results on both synthetic and real DT-MRI data.
TL;DR: A direct geometrical description of the time development of a Hamiltonian potential model and some examples of unstable Hamiltonian systems in two dimensions are discussed.
Abstract: The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model when a transition is made to an associated manifold. We find, in this way, a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this associated manifold results in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions.