The Ricci flow as a geodesic on the manifold of Riemannian metrics

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.     

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L ² Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L ² metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.     

Spaces of Riemannian metrics

In this paper, we consider spaces M of Riemannian metrics on a closed manifold M. In the case where the manifold M is equipped with a symplectic or contact structure, we consider spaces AM of associated metrics. We study geometric and topological properties of these spaces and Riemannian functionals on spaces of metrics. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 31, Geometry, 2005.     

The Riemannian <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> topology on the manifold of Riemannian metrics

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L ² Riemannian metric—so-called because it induces an L ² topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L ¹-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L ² metric. We also give a user-friendly criterion for convergence (with respect to the L ² metric) in the manifold of metrics.     

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.     

Existence of Dirac eigenvalues of higher multiplicity

In this article, we prove that on any compact spin manifold of dimension \(m \equiv 0,6,7 \mod 8\), there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by “catching” the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.     

Maximally movable spaces of Finsler type and their generalization

In this paper, we consider some generalization of maximally movable spaces of Finsler type. Among them, there are locally conic spaces (Riemannian metrics of their tangent spaces are realized on circular cones) and generalized Lagrange spaces with Tamm metrics (their tangent Riemannian spaces admit all rotations). On the tangent bundle of a Riemannian manifold, we study a special class of almost product metrics, generated Tamm metric. This class contains Sasaki metric and Cheeger–Gromol metric. We determine the position of this class in the Naveira classification of Riemannian almost product metrics.     

A converse of the Gelfand theorem

In this short note we obtain a converse to the Gelfand theorem: a Riemannian manifold is homogeneous if the isometrically invariant operators on the manifold form a commutative algebra.

     

A remark on the Harnack inequality for non-self-adjoint evolution equations

In this paper we consider a non-self-adjoint evolution equation on a compact Riemannian manifold with boundary. We prove a Harnack inequality for a positive solution satisfying the Neumann boundary condition. In particular, the boundary of the manifold may be nonconvex and this gives a generalization to a theorem of Yau.

     

A finiteness theorem for harmonic maps into Hilbert Grassmannians

In this article we demonstrate that every harmonic map from a closed Riemannian manifold into a Hilbert Grassmannian has image contained within a finite-dimensional Grassmannian.

     

Finsler manifolds with nonpositive flag curvature and constant S-curvature

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in Rⁿ with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler metric with negative flag curvature and constant S-curvature must be Riemannian if the manifold is compact. We also study the nonpositive flag curvature case.supported by the National Natural Science Foundation of China (10371138).     

Geodesics of Sasakian Metrics on Tensor Bundles

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds. Dedicated to the memory of Vladimir Vishnevskii (1929-2007)     

Quadratic Interpolation on Spheres

Riemannian quadratics are C ¹ curves on Riemannian manifolds, obtained by performing the quadratic recursive deCastlejeau algorithm in a Riemannian setting. They are of interest for interpolation problems in Riemannian manifolds, such as trajectory-planning for rigid body motion. Some interpolation properties of Riemannian quadratics are analysed when the ambient manifold is a sphere or projective space, with the usual Riemannian metrics.     

Degenerate Riemannian metrics

We give a criterion for the existence of a degenerate Riemannian metric on a paracompact C^∞ manifold. We give a criterion for the existence of a corresponding connection without torsion on manifolds with degenerate Riemannian metrics.     

Riemannian metrics on ℝ<Superscript> <Emphasis Type="Italic">n</Emphasis> </Superscript> and Sobolev-type Inequalities

Poincaré-type estimates for a logarithmically concave measure μ on a convex set Ω are obtained. For this purpose, Ω is endowed with a Riemannian metric g in which the Riemannian manifold with measure (Ω, g, μ) has nonnegative Bakry–Emery tensor and, as a corollary, satisfies the Brascamp–Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincare, Hardy, or log-Sobolev type inequalities and other results.     

A quasifibration of spaces of positive scalar curvature metrics

In this paper we show that for Riemannian manifolds with boundary the natural restriction map is a quasifibration between spaces of metrics of positive scalar curvature. We apply this result to study homotopy properties of spaces of such metrics on manifolds with boundary.

     

A characterization of Riemannian flows

We prove that a flow on a closed manifold is Riemannian if and only if it is locally generated by Killing vector fields for a Riemannian metric.

     

Einstein condition and twistor spaces of compatible partially complex structures

We study the Einstein condition for a natural family of Riemannian metrics on the twistor space of partially complex structures of a fixed rank on the tangent spaces of a Riemannian manifold compatible with its metric. A generalization of the Einstein condition (discussed in the Besse book [Enstein Manifolds, Ergeb. Math. Grensgeb. (3), vol. 10, Springer, New York, 1987]) is also considered.     

Harmonic morphisms as unit normal bundles¶of minimal surfaces

Let be an isometric immersion between Riemannian manifolds and be the unit normal bundle of f. We discuss two natural Riemannian metrics on the total space and necessary and sufficient conditions on f for the projection map to be a harmonic morphism. We show that the projection map of the unit normal bundle of a minimal surface in a Riemannian manifold is a harmonic morphism with totally geodesic fibres. Received: 6 February 1999     

On the non-Riemannian quantity H of a Finsler metric

One of fundamental problems in Finsler geometry is to establish some delicate equations between Riemannian invariants and non-Riemannian invariants. Inspired by results due to Akbar-Zadeh etc., this note establishes a new fundamental equation between non-Riemannian quantity H and Riemannian quantities on a Finsler manifold. As its application, we show that all R-quadratic Finsler metrics have vanishing non-Riemannian invariant H generalizing result previously only known in the case of Randers metric.