Invariant Metrics Research Articles

On Isometries of Conformally Invariant Metrics

We prove that isometries in a conformally invariant metric of a general domain are quasiconformal. In the particular case of the punctured space, we prove that isometries in this metric are Möbius, thus resolving a conjecture of Ferrand et al. (J Anal Math 56: 187–120, 1991) in this particular case.

Asymptotic property C of the countable direct sum of the integers

It is proved that the countable direct sum of the integers with a proper invariant metric has asymptotic property C, which answers a question posed by A. Dranishnikov and M. Zarichnyi.

Indefinite metric of R. Mrugala and the geometry of thermodynamical phase space

We study an indefinite metric G which was introduced by R. Mrugala and is defined on the contact phase space (P,?) of a homogeneous thermodynamical system. We describe the curvature properties and the isometry group of the metric G. We established an isomorphism of the space (P,?,G) with the Heisenberg Lie group Hn, endowed with the right invariant contact structure and the right invariant indefinite metric. The lift of the metric G to the symplectization of contact space (P,?) and its properties are studied. Finally we introduce the hyperbolic projectivization of the space () that can be considered as the natural compactification of the thermodynamical phase space (P, q, G).

$$G$$ G -monopole invariants on some connected sums of 4-manifolds

On a smooth closed oriented 4-manifold \(M\) with a smooth action of a finite group \(G\) on a Spin\(^c\) structure, \(G\)-monopole invariant is defined by “counting” \(G\)-invariant solutions of Seiberg–Witten equations for any \(G\)-invariant Riemannian metric on \(M\). We compute \(G\)-monopole invariants on some \(G\)-manifolds. For example, the connected sum of \(k\) copies of a 4-manifold with nontrivial mod 2 Seiberg–Witten invariant has nonzero \(\mathbb Z_k\)-monopole invariant mod 2, where the \(\mathbb Z_k\)-action is given by cyclic permutations of \(k\) summands.

Invariant metrics on Lie groups

Index formulas for the curvature tensors of an invariant metric on a Lie group are obtained. The results are applied to the problem of characterizing invariant metrics of zero and non-zero constant curvature. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics.

Quantification of microscopic anisotropy with diffusion MRI

Quantification of microscopic anisotropy with diffusion MRI

Geodesic vectors and subalgebras in two-step nilpotent metric Lie algebras

Abstract A metric Lie algebra is a Lie algebra endowed with a Euclidean inner product. A subalgebra is called flat, respectively totally geodesic, if its exponential image in the corresponding Lie group with left invariant Riemannian metric is flat, respectively a totally geodesic submanifold. A non-zero vector is geodesic, if the generated one-dimensional subspace is totally geodesic. We study geodesic vectors and flat totally geodesic subalgebras in two-step nilpotent metric Lie algebras and show that their linear structure is independent of the inner product of the metric Lie algebra. We determine the geodesic vectors and the flat totally geodesic subalgebras in the two-step nilpotent metric Lie algebras of dimension ≤ 6.

First passage percolation on nilpotent Cayley graphs and beyond

We prove an asymptotic shape theorem for first-passage percolation on Cayley graphs of virtually nilpotent groups. By a theorem of Pansu, the asymptotic cone of a finitely generated nilpotent group is isometric to a simply connected nilpotent Lie group equipped with some left invariant Carnot-Caratheodory metric. Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a companion work by the second author, the proof relies on Talagrand's concentration inequality, and on Pansu's theorem. Adapting an argument of Benjamini, Kalai and Schramm, we prove a sublinear estimate on the variance for virtually nilpotent groups which are not virtually isomorphic to Z. We further discuss the asymptotic cones of first-passage percolation on general infinite connected graphs: we prove that the asymptotic cones are a.e. deterministic if and only the volume growth is subexponential.

Invariants of complex structures on nilmanifolds

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [7], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants. We show how to use a Riemannian invariant: the eigenvalues of the Ricci operator, polynomial invariants and discrete invariants to give an alternative proof of the pairwise non-isomorphism between the structures which have appeared in the classification of abelian complex structures on 6-dimensional nilpotent Lie algebras given in [1]. We also present some continuous families in dimension 8.

On the degeneration of some 3-manifold geometries via unit groups of quaternion algebras

A two parameter continuous family of three-dimensional Lie groups with a left invariant Riemannian metric is defined. Each of these Lie groups is the unit group of a quaternion algebra. All the possible left invariant Riemannian structures in the Heisenberg group appear as limit cases. The degeneration of some Thurston’s 3-manifold geometries are studied in this framework. Among other interesting degenerations, the degeneration spherical-Nil-\(\widetilde{SL(2,\mathbb {R})}\) is obtained.

Contact 3-manifolds with the Reeb-flow symmetry

We prove that the Ricci operator on a contact Riemannian 3-manifold $M$ is invariant along the Reeb flow if and only if $M$ is Sasakian or locally isometric to $\mathrm{SU}(2)$ (or $\mathrm{SO}(3)$), $\mathrm{SL}(2,\boldsymbol{R})$ (or $O(1,2)$), the group $E(2)$ of rigid motions of Euclidean 2-plane with a contact left invariant Riemannian metric.

Yang-Mills gauge fields conserving the symmetry algebra of the Dirac equation in a homogeneous space

We consider the Dirac equation with an external Yang-Mills gauge field in a homogeneous space with an invariant metric. The Yang-Mills fields for which the motion group of the space serves as the symmetry group for the Dirac equation are found by comparison of the Dirac equation with an invariant matrix differential operator of the first order. General constructions are illustrated by the example of de Sitter space. The eigenfunctions and the corresponding eigenvalues for the Dirac equation are obtained in the space R2 × S2 by a noncommutative integration method.

Einstein equations for invariant metrics on flag spaces and their Newton polytopes

Einstein equations for invariant metrics on flag spaces and their Newton polytopes

Sub-Riemannian Geometry on Infinite-Dimensional Manifolds

We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $$M$$ , the metric is defined only on a sub-bundle $${{\mathrm{\mathcal {H}}}}$$ of the tangent bundle $$TM$$ , called the horizontal distribution. Similarly to the finite-dimensional case, we are able to split possible candidates for minimizing curves into two categories: semi-rigid curves that depend only on $${{\mathrm{\mathcal {H}}}}$$ , and normal geodesics that depend both on $${{\mathrm{\mathcal {H}}}}$$ itself and on the metric on $${{\mathrm{\mathcal {H}}}}$$ . In this sense, semi-rigid curves in the infinite-dimensional case generalize the notion of singular curves for finite dimensions. In particular, we study the case of regular Lie groups with invariant sub-Riemannian structure. As examples, we consider the group of sense-preserving diffeomorphisms $${{\mathrm{Diff}}}\, S^1$$ of the unit circle and the Virasoro–Bott group with their respective horizontal distributions chosen to be the Ehresmann connections with respect to a projection to the space of normalized univalent functions. In these cases we prove controllability and find formulas for the normal geodesics with respect to the pullback of the invariant Kahlerian metric on the class of normalized univalent functions. The geodesic equations are analogues to the Camassa–Holm, Hunter–Saxton, KdV, and other known non-linear PDE.

The left invariant metric in the general linear group

Left invariant metrics induced by the p -norms of the trace in the matrix algebra are studied on the general linear group. By means of the Euler–Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p = 2 , are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p -summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.

Milin's coefficients, complex geometry of Teichmüller spaces and variational calculus for univalent functions

Abstract We investigate the invariant metrics and complex geodesics in the universal Teichmüller space and the Teichmüller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all non-expanding invariant metrics in either of these spaces coincide with its intrinsic Teichmüller metric. Other applications concern the variational theory for univalent functions with quasiconformal extension. It turns out that geometric features caused by the equality of metrics and connection with complex geodesics provide deep distortion results for various classes of such functions and create new phenomena which do not appear in the classical geometric function theory.

Affine hypersurfaces with parallel difference tensor relative to affineα-connection

Li and Zhang (2014) studied affine hypersurfaces of R n + 1 with parallel difference tensor relative to the affine α -connection ∇ ( α ) , and characterized the generalized Cayley hypersurfaces by K n − 1 ≠ 0 and ∇ ( α ) K = 0 for some nonzero constant α , where the affine α -connection ∇ ( α ) of information geometry was introduced on affine hypersurface. In this paper, by a slightly different method we continue to study affine hypersurfaces with ∇ ( α ) K = 0 , if α = 0 we further assume that the Pick invariant vanishes and affine metric is of constant sectional curvature. It is proved that they are either hyperquadrics or improper affine hypersphere with flat indefinite affine metric, the latter can be locally given as a graph of a polynomial of at most degree n + 1 with constant Hessian determinant. In particular, if the affine metric is definite, Lorentzian, or its negative index is 2, we complete the classification of such hypersurfaces.

Left invariant Randers metrics on the 3-dimensional Heisenberg group

In the previous paper [MR2430243] we computed some geometric quantities such as curvature and flag curvature for a general left invariant Finsler metric on a two-step nilpotent group. In the present paper we give a more complete description of the Chern--Rund connection defined by a left invariant Randers metric on the 3 dimensional Heisenberg group.

Graev metrics on free products and HNN extensions

We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev’s construction of metrics on free groups over pointed metric spaces.

Homogeneous Einstein Metrics on Certain Generalized Flag Manifolds with Six Isotropy Summands

For a generalized flag manifold M = G/K of a compact connected simple Lie group G whose isotropy representation decomposes into more than five isotropy summands, there are only a few results about the homogeneous Einstein metrics on M. Finding the invariant Einstein metrics on generalized flag manifolds, there are two difficulties. One is computing the non-zero structure constants, the other is computing the Grobner basis of the system of Einstein equations. In this paper, we give a method (Theorem A) which can be used to calculate structure constants of generalized flag manifolds with any number of isotropy summands. In this direction we present invariant Einstein metrics on some flag manifolds of exceptional groups with six isotropy summands.