Invariant Riemannian Metrics Research Articles

Geodesics, Parallel Transport & One-Parameter Subgroups for Diffeomorphic Image Registration

Computational anatomy aims at developing models to understand the anatomical variability of organs and tissues. A widely used and validated instrument for comparing the anatomy in medical images is non-linear diffeomorphic registration which is based on a rich mathematical background. For instance, the large deformation diffeomorphic metric mapping (LDDMM) framework defines a Riemannian setting by providing a right invariant metric on the tangent spaces, and solves the registration problem by computing geodesics parametrized by time-varying velocity fields. A simpler alternative based on stationary velocity fields (SVF) has been proposed, using the one-parameter subgroups from Lie groups theory. In spite of its better computational efficiency, the geometrical setting of the SVF is more vague, especially regarding the relationship between one-parameter subgroups and geodesics. In this work, we detail the properties of finite dimensional Lie groups that highlight the geometric foundations of one-parameter subgroups. We show that one can define a proper underlying geometric structure (an affine manifold) based on the canonical Cartan connections, for which one-parameter subgroups and their translations are geodesics. This geometric structure is perfectly compatible with all the group operations (left, right composition and inversion), contrarily to left- (or right-) invariant Riemannian metrics. Moreover, we derive closed-form expressions for the parallel transport. Then, we investigate the generalization of such properties to infinite dimensional Lie groups. We suggest that some of the theoretical objections might actually be ruled out by the practical implementation of both the LDDMM and the SVF frameworks for image registration. This leads us to a more practical study comparing the parameterization (initial velocity field) of metric and Cartan geodesics in the specific optimization context of longitudinal and inter-subject image registration.Our experimental results suggests that stationarity is a good approximation for longitudinal deformations, while metric geodesics notably differ from stationary ones for inter-subject registration, which involves much larger and non-physical deformations. Then, we turn to the practical comparison of five parallel transport techniques along one-parameter subgroups. Our results point out the fundamental role played by the numerical implementation, which may hide the theoretical differences between the different schemes. Interestingly, even if the parallel transport generally depends on the path used, an experiment comparing the Cartan parallel transport along the one-parameter subgroup and the LDDMM (metric) geodesics from inter-subject registration suggests that our parallel transport methods are not so sensitive to the path.

Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H, K, G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachhöfer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.

Nonnegatively curved homogeneous metrics in low dimensions

We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups $(H,K,G)$ whether nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachh\"ofer and K. Tapp \cite{ST}, we examine all $G$-invariant fibration metrics on $G/H$ for $G$ a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of nonnegative sectional curvature.

RICCI AND SCALAR CURVATURES ON SU(3)

In this paper, we obtain the Ricci curvature and the scalar curvature on SU(3) with some left invariant Riemannian metric. And then we get a necessary and sufficient condition for the scalar curvature (resp. the Ricci curvature) on the Riemannian manifold SU(3) to be positive.

HARMONIC MAPS BETWEEN THE GROUP OF AUTOMORPHISMS OF THE QUATERNION ALGEBRA

In this paper, let Q be the real quaternion algebra which consists of all quaternionic numbers, and let G be the Lie group of all automorphisms of the algebra Q. Assume that g is an arbitrary given left invariant Riemannian metric on the Lie group G. Then, we obtain a necessary and sufficient condition for an automorphism of the group G to be harmonic.

The isometry groups of simply connected 3-dimensional unimodular Lie groups

For each simply connected 3-dimensional unimodular Lie group the left invariant Riemannian metrics up to automorphism are classified in Ha and Lee (2009) [2]. Using this classification, we determine on each such a Lie group the full group of isometries which depends on left invariant Riemannian metrics.

Filiform nilsolitons of dimension 8

A Riemannian manifold (M, g) is said to be Einstein if its Ricci tensor satisfies ric (g) = cg, for some c is an element of R. In the homogeneous case, a problem that is still open is the so-called Alekseevskii conjecture. This conjecture says that any homogeneous Einstein space with negative scalar curvature (i.e., c < 0) is a solvmanifold: a simply connected solvable Lie group endowed with a left invariant Riemannian metric. The aim of this paper is to classify Einstein solvmanifolds of dimension 9 whose nilradicals are filiform (i.e., (n - 1)-step nilpotent and n-dimensional).

Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic

The authors give a short survey of previous results on generalized normal homogeneous ( δ-homogeneous, in other terms) Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with nonnegative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these results, they prove that the family of all compact simply connected indecomposable generalized normal homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds Sp ( l ) / U ( 1 ) ⋅ Sp ( l − 1 ) = C P 2 l − 1 , l ⩾ 2 , supplied with invariant Riemannian metrics of positive sectional curvature with the pinching constants (the ratio of the minimal sectional curvature to the maximal one) in the open interval ( 1 / 16 , 1 / 4 ) . This implies very unusual geometric properties of the adjoint representation of Sp ( l ) , l ⩾ 2 . Some unsolved questions are suggested.

On Euler–Arnold equations and totally geodesic subgroups

The geodesic motion on a Lie group equipped with a left or right invariant Riemannian metric is governed by the Euler–Arnold equation. This paper investigates conditions on the metric in order for a given subgroup to be totally geodesic. Results on the construction and characterisation of such metrics are given, especially in the special case of easy totally geodesic submanifolds that we introduce. The setting works both in the classical finite dimensional case, and in the category of infinite dimensional Fréchet–Lie groups, in which diffeomorphism groups are included. Using the framework we give new examples of both finite and infinite dimensional totally geodesic subgroups. In particular, based on the cross helicity, we construct right invariant metrics such that a given subgroup of exact volume preserving diffeomorphisms is totally geodesic. The paper also gives a general framework for the representation of Euler–Arnold equations in arbitrary choice of dual pairing.

Todd's maximum-volume ellipsoid problem on symmetric cones

Let V be a Euclidean Jordan algebra and let Ω be the associated symmetric cone, a self-dual homogeneous open convex cone, which is a symmetric space of noncompact type under G(Ω) (the linear automorphism group)-invariant Riemannian metric. We show that the radius of the largest ball centered at a ∈ Ω inscribed in Ω coincides with its minimum eigenvalue and then provide a proof of the problem of finding a point x ∈ Ω to maximize the product of the radii of the largest balls centered at a, b ∈ Ω and inscribed in Ω of the tangent space Tx (Ω) and its dual space Tx–1 (Ω), respectively. We obtain an explicit formula for the maxima; it is precisely the minimal eigenvalue of P (a1/2)b where P denotes the quadratic representation of V. This provides an affirmative answer to a question of Todd on the maxima.

The Relation Between Automorphism Group and Isometry Group of Randers Lie Groups

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invariant vector field, in a special case, the Lie group admits a left invariant Randers metric such that this intersection is a maximal compact subgroup of the group of automorphisms with respect to which the considered vector field is invariant.

Examples of homogeneous vectors on the fiber space of the associated and the tangent bundles

The paper considers the associated bundle ξ = (G × KG/K, ρξ, G/K, G/K) and the tangent bundle τG/K = (TG/K, πG/K, G/K, Rm), and gives special examples of odd dimensional solvable Lie groups equipped with left invariant Riemannian metric. Some conditions about existence of homogeneous geodesic vectors on the fiber space of ξ and τG/K are proved.

Left invariant metrics and curvatures on simply connected three‐dimensional Lie groups

AbstractFor each simply connected three‐dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the principal Ricci curvatures, the scalar curvature and the sectional curvatures as functions of left invariant metrics on the three‐dimensional Lie groups. Our results improve a bit of Milnor's results of [7] in the three‐dimensional case, and Kowalski and Nikv́cević's results [6, Theorems 3.1 and 4.1] (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Multiplicative noncommutative deformations of left invariant Riemannian metrics on Heisenberg groups

Let H n be the Heisenberg group of dimension 2 n + 1 . We give a precise description of all ( π , 〈 , 〉 ) , where π is a multiplicative Poisson tensor on H n and 〈 , 〉 is a left invariant metric on H n such that ( π , 〈 , 〉 ) satisfies the necessary conditions, introduced by Eli Hawkins, to the existence of a noncommutative deformation of the spectral triple associated to ( H n , 〈 , 〉 ) . To cite this article: A. Bahayou, M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Some Berwald spaces of non-positive flag curvature

In this paper, by using left invariant Riemannian metrics on some three-dimensional Lie groups, we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally Minkowskian spaces of zero constant flag curvature.

On the Flag Curvature of Invariant Randers Metrics

In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field.

Regularization of diffusion tensor MRI via local coordinates

AbstractWe propose a novel framework for regularization of symmetric positive‐definite (SPD) tensors (e.g., diffusion tensors). This framework is based on differential geometry. The space of SPD matrices, Pn, is described as a Riemannian manifold that is parameterized via the Iwasawa coordinate system. Then, distances on Pn are measured in terms of a natural GL (n)‐invariant Riemannian metric. Using the Beltrami framework we construct a set of coupled geometric PDEs with respect to the Iwasawa coordinates. Then, by means of the gradient descent method these equations define the regularization flow over Pn. It appears to be that the local coordinate approach via that coordinate system results in very simple numerics that leads to fast convergence of the algorithm. We demonstrate the efficiency of this algorithm on real volumetric DTI datasets. Results of fibers tractography before and afterthe regularization process arepresented. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

UNIMODULAR GROUPS OF TYPE ℝ3⋊ ℝ

There are 7 types of 4-dimensional solvable Lie groups of the form <TEX>${\mathbb{R}^3}\;{\times}_{\varphi}\;{\mathbb{R}}$</TEX> which are unimodular and of type (R). They will have left. invariant Riemannian metrics with maximal symmetries. Among them, three nilpotent groups <TEX>$({\mathbb{R}^4},\;Nil^3\;{\times}\;{\mathbb{R}\;and\;Nil^4)$</TEX> are well known to have lattices. All the compact forms modeled on the remaining four solvable groups <TEX>$Sol^3\;{\times}\;{\mathbb{R}},\;Sol_0^4,\;Sol_0^'4\;and\;Sol_{\lambda}^4$</TEX> are characterized: (1) <TEX>$Sol^3\;{\times}\;{\mathbb{R}}$</TEX> has lattices. For each lattice, there are infra-solvmanifolds with holonomy groups 1, <TEX>${\mathbb{Z}}_2\;or\;{\mathbb{Z}}_4$</TEX>. (2) Only some of <TEX>$Sol_{\lambda}^4$</TEX>, called <TEX>$Sol_{m,n}^4$</TEX>, have lattices with no non-trivial infra-solvmanifolds. (3) <TEX>$Sol_0^{'4}$</TEX> does not have a lattice nor a compact form. (4) <TEX>$Sol_0^4$</TEX> does not have a lattice, but has infinitely many compact forms. Thus the first Bieberbach theorem fails on <TEX>$Sol_0^4$</TEX>. This is the lowest dimensional such example. None of these compact forms has non-trivial infra-solvmanifolds.

U(n + 1) × U(p + 1)-Hermitian metrics on the manifold S 2n+1 × S 2p+1

A two-parameter family of invariant almost-complex structures J α,c is given on the homogeneous space M × M’ = U(n + 1)/U(n) × U(p + 1)/U(p); all these structures are integrable. We consider all invariant Riemannian metrics on the homogeneous space M × M’. They depend on five parameters and are Hermitian with respect to some complex structure J α,c . In this paper, we calculate the Ricci tensor, scalar curvature, and obtain estimates of the sectional curvature for any metric on M × M’. All the invariant metrics of nonnegative curvature are described. We obtain the extremal values of the scalar curvature functional on the four-parameter family of metrics g α,c,λ,λ’;1 .

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ 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. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants. Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.