Canonical Riemannian Metric Research Articles

Moufang patterns and geometry of information

Technology of data collection and transmission is based on various mathematical models of encoding. The words Geometry of information refer to such models, whereas the words patterns refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.

Four-dimensional closed manifolds admit a weak harmonic Weyl metric

On four-dimensional closed manifolds we introduce a class of canonical Riemannian metrics, that we call weak harmonic Weyl metrics, defined as critical points in the conformal class of a quadratic functional involving the norm of the divergence of the Weyl tensor. This class includes Einstein and, more in general, harmonic Weyl manifolds. We prove that every closed four-manifold admits a weak harmonic Weyl metric, which is the unique (up to dilations) minimizer of the corresponding functional in a suitable conformal class. In general the problem is degenerate elliptic due to possible vanishing of the Weyl tensor. In order to overcome this issue, we minimize the functional in the conformal class determined by a reference metric, constructed by Aubin, with nowhere vanishing Weyl tensor. Moreover, we show that anti-self-dual metrics with positive Yamabe invariant can be characterized by pinching conditions involving suitable quadratic Riemannian functionals.

Conformal transformations of the pseudo‐Riemannian metric of a homogeneous pair

We introduce a new notion of a homogeneous pair for a pseudo-Riemannian metric $g$ and a positive function $f$ on a manifold $M$ admitting a free $\mathbb{R}_{>0}$-action. There are many examples admitting this structure. For example, (a) a class of pseudo-Hessian manifolds admitting a free $\mathbb{R}_{>0}$-action and a homogeneous potential function such as the moduli space of torsion-free $G_2$-structures, (b) the space of Riemannian metrics on a compact manifold, and (c) many moduli spaces of geometric structures such as torsion-free ${\rm Spin}(7)$-structures admit this structure. Hence we provide the unified method for the study of these geometric structures. We consider conformal transformations of the pseudo-Riemannian metric $g$ of a homogeneous pair $(g, f)$. Showing that the pseudo-Riemannian manifold $(M, (v \circ f) g)$, where $v: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ is a smooth function, has the structure of a warped product, we study the geometric structures such as the sectional curvature, geodesics and the metric completion (if $g$ is positive definite) w.r.t. $(v \circ f) g$ in terms of those on the level set of $f$. In particular, (1) we can generalize the result of Clarke and Rubinstein about the metric completion of the space of Riemannian metrics w.r.t. the conformal transformations of the Ebin metric, and (2) two canonical Riemannian metrics on the $G_2$ moduli space have different metric completions.

A potential generalization of some canonical Riemannian metrics

The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases, we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper, we also describe the “nongradient” version of this construction.

Generalization of Finsler metrics on the product of Finsler manifolds

In Riemannian manifolds, there exists a canonical Riemannian metric on the product of them (Lee in Riemannian geometry an introduction to curvature, Springer, New York, 1997). But at the product of Finsler manifolds, the canonical Finsler metric has not been defined. In this paper, we are going to study the product of Finsler manifolds and give a canonical Finsler metric on it.

Isometry types of frame bundles

We consider the orthonormal frame bundle F(M) of a Riemannian manifold M. A construction of Sasaki defines a canonical Riemannian metric on F(M). We prove that for two closed Riemannian n-manifolds M and N, the frame bundles F(M) and F(N) are isometric if and only if M and N are isometric, except possibly in dimensions 3, 4, and 8. This answers a question of Benson Farb except in dimensions 3, 4, and 8.

Submanifolds with homothetic Gauss map in codimension two

Let \(f{:}\;M^n\rightarrow \mathbb {R}^{n+p}\) be an isometric immersion of an n-dimensional Riemannian manifold \(M^n\) into the (\(n+p\))-dimensional Euclidean space. Its Gauss map \(\phi {:}\;M^n\rightarrow G_n(\mathbb {R}^{n+p})\) into the Grassmannian \(G_n(\mathbb {R}^{n+p})\) is defined by assigning to every point of \(M^n\) its tangent space, considered as a vector subspace of \(\mathbb {R}^{n+p}\). The third fundamental form \(\text{ III }\) of f is the pullback of the canonical Riemannian metric on \(G_p(\mathbb {R}^{n+p})\) via \(\phi \). In this article we derive a complete classification of all those f with codimension two for which the Gauss map \(\phi \) is homothetic; i.e., \(\text{ III }\) is a constant multiple of the Riemannian metric on \(M^n\). We furthermore study and classify codimension two submanifolds with homothetic Gauss map in real space forms of nonzero curvature. To conclude, based on a strong connection established between homothetic Gauss map and minimal Einstein submanifolds, we pose a conjecture suggesting a possible complete classification of the submanifolds with the former property in arbitrary codimension.

COMPUTING CONSTANT-CURVATURE METRICS FOR HYPERBOLIC 3-MANIFOLDS WITH BOUNDARIES USING TRUNCATED TETRAHEDRAL MESHES

Every surface in the Euclidean space ℝ3 admits a canonical Riemannian metric that has constant Gaussian curvature and is conformal to the original metric. Similarly, 3-manifolds can be decomposed into pieces that admit canonical metrics. Such metrics not only have theoretical significance in 3-manifold geometry and topology, but also have potential applications to practical problems in engineering fields such as shape classification.In this paper we present an algorithm that is based on a discrete curvature flow to compute constant curvature metrics on 3-manifolds that are hyperbolic and have boundaries of a certain type. We also provide an approach to visualize such a metric by embedding the fundamental domain and universal covering in the hyperbolic space ℍ3. Some experimental results are given for both algorithms.Furthermore, we propose an algorithm to automatically construct truncated tetrahedral meshes for 3-manifolds with boundaries. It can not only generate inputs to the curvature flow algorithm, but could also serve as an automatic tool for geometers and topologists to build simple models for complicated 3-manifolds, and therefore facilitate their research that requires such models.

On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Let M =G/H be an irreducible homogeneous compact manifold of dimension n equipped with its canonical Riemannian metric. Let γ be the lowest nonzero eigenvalue of the Laplace operator. Let μ be the normalized Haar measure and μt be the heat diffusion measure, i.e., the law of Brownian motion started at a fixed origin in M. We show that the total variation distance between μt and μ is not small for t ≪λ−1 logn.This is sharp, up to a factor of two, in the case of compact irreducible simply connected symmetric spaces.

Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

A.A. Kirillov has given a parametrization of the space U ∞ of univalent functions on the closed unit disk, which are C ∞ up to the boundary, by Diff ( S 1 ) / S 1 where Diff ( S 1 ) denotes the group of orientation preserving diffeomorphisms of the circle S 1 . In the same spirit, the space J ∞ of C ∞ Jordan curves in the complex plane can be parametrized by the double quotient SU ( 1 , 1 ) \\ Diff ( S 1 ) / SU ( 1 , 1 ) . As a consequence, J ∞ carries a canonical Riemannian metric. We construct a canonical Brownian motion on U ∞ . Classical technologies of the theory of univalent functions, like Beurling–Ahlfors extension, Loewner equation, Beltrami equation, developed in the context of Kunita's stochastic flows, are the tools for obtaining this result which should be seen as a first step to the construction of a canonical Brownian motion on J ∞ .

Affine spheres and convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-manifolds

We outline consequences of a theorem of Cheng-Yau in affine differential geometry for manifolds locally modeled on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. In particular, for properly convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]-manifolds, we observe that there are two canonical projectively flat connections and a canonical Riemannian metric. One connection represents the given [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]-structure and the other the [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]-structure of the projective dual manifold. When n is 2, we use this approach together with a result of C.-P. Wang to show that a compact oriented convex [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /]-surface of genus at least two is equivalent to a conformal structure on the surface together with a holomorphic tricanonical form. This recovers a theorem of Goldman on the deformation space of such surfaces, and yields a description of the moduli space.

Standard pseudo-Hermitian structure on manifolds and seifert fibration

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function Λ onM. AsM supports a contact form, there exists a characteristic vector field ξ dual to the contact structure. If ξ induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature Λ or of nonpositive curvature Λ. By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature Λ turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature Λ (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When Λ≤0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.

Isometric immersions with homothetical Gauss map

Let f:M→ℝ be an isometric immersion of an m-dimensional Riemannian manifold M into the n-dimensional Euclidean space. Its Gauss map g:M→G m (ℝ n ) into the Grassmannian G m (ℝ n ) is defined by assigning to every point of M its tangent space, considered as a vector subspace of ℝ n . The third fundamental form b of f is the pull-back of the canonical Riemannian metric on G m (ℝ n ) via g. In this article we derive a complete classification of all those f (with flat normal bundle) for which the Gauss map g is homothetical; i.e. b is a constant multiple of the Riemannian metric on M. Using these results we furthermore classify all those f (with flat normal bundle) for which the third fundamental form b is parallel w.r.t. the Levi-Civita connection on M.

A metric on the manifold of immersions and its Riemannian curvature

E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n−1) dimensional manifold into ℝn, and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.

On the volume of the generalized Gauss map

Let f :M~ N"+P be an immersion of the n-manifold M into Euclidean (n + p)-space. The classical Gauss map for surfaces in 3-space has two natural generalizations to this situation. In the obvious way we can assign (a) to every unit normal vector of f a point in the unit sphere S "+p- 1, or (b) to every point of M its p-dimensional normal space through tile origin of ~"+P. Both the sphere S "+p- 1 and the Grassmannian Gv(~ "+p) have canonical Riemannian metrics, and we can talk about the (n + p - 1)-dimensional or the n-dimensional volumes of the Gauss maps (a) and (b). In the first case this (suitably normalized) volume is the well-known total absolute curvature r(f) off. The Gauss map (b) has also been studied extensively (see [8] for a survey and a list of references). In the present note we shall affirmatively answer a conjecture of Osserman [8] stating that the (again suitably normalized) volume a(f) of the map (b) satisfies z(f) 0 denote the volume of the unit k-sphere, and define

Eigenvalues of a Laplacian and commutative Lie subalgebras

If K is a compact semi-simple Lie group and g is the complexification of its Lie algebra then one knows that the algebra ? of (Maurer-Cartan) complex-valued left invariant differential forms may be naturally identified with the exterior algebra ?g. Also, one knows then that ?g is stable under the Laplacian defined with respect to the canonical Riemannian metric on K.