Riemannian Invariants Research Articles

Log-Determinant Divergences Revisited: Alpha-Beta and Gamma Log-Det Divergences

This work reviews and extends a family of log-determinant (log-det) divergences for symmetric positive definite (SPD) matrices and discusses their fundamental properties. We show how to use parameterized Alpha-Beta (AB) and Gamma log-det divergences to generate many well-known divergences; in particular, we consider the Stein’s loss, the S-divergence, also called Jensen-Bregman LogDet (JBLD) divergence, Logdet Zero (Bhattacharyya) divergence, Affine Invariant Riemannian Metric (AIRM), and other divergences. Moreover, we establish links and correspondences between log-det divergences and visualise them on an alpha-beta plane for various sets of parameters. We use this unifying framework to interpret and extend existing similarity measures for semidefinite covariance matrices in finite-dimensional Reproducing Kernel Hilbert Spaces (RKHS). This paper also shows how the Alpha-Beta family of log-det divergences relates to the divergences of multivariate and multilinear normal distributions. Closed form formulas are derived for Gamma divergences of two multivariate Gaussian densities; the special cases of the Kullback-Leibler, Bhattacharyya, Rényi, and Cauchy-Schwartz divergences are discussed. Symmetrized versions of log-det divergences are also considered and briefly reviewed. Finally, a class of divergences is extended to multiway divergences for separable covariance (or precision) matrices.

Lessons fromf(R,Rc2,Rm2,Lm)gravity: Smooth Gauss-Bonnet limit, energy-momentum conservation, and nonminimal coupling

This paper studies a generic fourth-order theory of gravity with Lagrangian density $f(R,R_c^2,R_m^2, \mathscr{L}_m)$. By considering explicit $R^2$ dependence and imposing the "coherence condition" $f_{R^2}\!=\!f_{R_m^2}\!=\! -f_{R_c^2}/4$, the field equations of $f(R,R^2,R_c^2,R_m^2, \mathscr{L}_m)$ gravity can be smoothly reduced to that of $f(R,\mathcal{G},\mathscr{L}_m)$ generalized Gauss-Bonnet gravity. We use Noether's conservation law to study the $f(\mathcal{R}_1,\mathcal{R}_2\ldots,\mathcal{R}_n,\mathscr{L}_m)$ model with nonminimal coupling between $\mathscr{L}_m$ and Riemannian invariants $ \mathcal{R}_i$, and conjecture that the gradient of nonminimal gravitational coupling strength $\nabla^\mu f_{\!\mathscr{L}_m}$ is the only source for energy-momentum non-conservation. This conjecture is applied to the $f(R,R_c^2,R_m^2, \mathscr{L}_m)$ model, and the equations of continuity and non-geodesic motion of different matter contents are investigated. Finally, the field equation for Lagrangians including the traceless-Ricci square and traceless-Riemann (Weyl) square invariants is derived, the $f(R,R_c^2,R_m^2, \mathscr{L}_m)$ model is compared with the $f(R,R_c^2,R_m^2,T)+2\kappa \mathscr{L}_m$ model, and consequences of nonminimal coupling for black hole and wormhole physics are considered.

Homogeneous spaces with inner metric and with integrable invariant distributions

This paper is a survey of results (partly obtained by the authors) on homogeneous spaces of Lie groups \(G\) with a compact stabilizer subgroup \(H\), on which every \(G\)-invariant distribution is integrable. It is proved that the condition of integrability is necessary and sufficient for every invariant inner metric to be (holonomic) Finsler on such a space. As a corollary of the obtained results, we assert that the class of homogeneous spaces with invariant non-holonomic Riemannian metrics (in other terms, sub-Riemannian or Carnot–Caratheodory metrics), which were actively studied last 3 decades after Gromov’s work, is rather broad. On the other hand, the class of homogeneous spaces with integrable invariant distributions includes Cartan’s symmetric spaces as well as isotropy irreducible, in particular, strictly isotropy irreducible, homogeneous spaces, which have been classified in simply connected case in the papers by Wang and Ziller (respectively, by Manturov, Wolf and Kramer). Special attention is paid to the case, when the Lie groups \(G\) and \(H\) are connected. Then the integrability condition of the invariant distributions is equivalent to a purely algebraic condition, that for the Lie algebra \(h\) of the subgroup \(H\), any \(ad(h)\)-invariant vector subspace in the Lie algebra \(g\) of the Lie group \(G\) is a Lie subalgebra; such Lie subalgebra \(h\subset g\) is called a strong subalgebra. The first author proved that a simply connected and compact space \(G/H\) with this condition is isomorphic to a direct product of strictly isotropy irreducible homogeneous spaces. In line with this, the second author recently found several non-compact simply connected homogeneous spaces with this condition, which are not isomorphic to such direct products. These results are naturally related to the structure questions of a class of general homogeneous locally compact spaces with an inner metric. This class is exactly the closure in the Gromov–Hausdorff sense of the class of homogeneous manifolds with an inner metric. Any such manifold is isometric to some homogeneous manifold \(G/H\) with \(G\)-invariant (may be, non-holonomic) Finsler metric. The authors give fairly detailed survey of the existing methods of the search of geodesics, i.e., locally shortest arcs, on such manifolds (particularly, with invariant non-holonomic Riemannian metrics), non-holonomic metric geometry and its relations with the geometric group theory, \(CR\)-manifolds, thermodynamics, etc. Some unsolved problems are suggested.

$R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms. We find isometries (called $R$-transforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We also show how to utilise the isometries to compute geodesics numerically.

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres \({S^n}\). We prove that for any connected (almost effective) transitive on \(S^n\) compact Lie group \(G\), the family of \(G\)-invariant Riemannian metrics on \(S^n\) contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and \(n\ge 5\). Any such family (that exists only for \(n=2k+1\)) contains a metric \(g_\mathrm{can}\) of constant sectional curvature \(1\) on \(S^n\). We also prove that \((S^{2k+1}, g_\mathrm{can})\) is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to \(G\) (except the groups \(G={ SU}(k+1)\) with odd \(k+1\)). The space of unit Killing vector fields on \((S^{2k+1}, g_\mathrm{can})\) from Lie algebra \(\mathfrak g \) of Lie group \(G\) is described as some symmetric space (except the case \(G=U(k+1)\) when one obtains the union of all complex Grassmannians in \(\mathbb{C }^{k+1}\)).

Certain Riemannian invariants for Sasakian submanifolds

Certain Riemannian invariants for Sasakian submanifolds

Thermodynamic curvature: pure fluids to black holes

Thermodynamics unavoidably contains fluctuation theory, expressible in terms of a unique thermodynamic information metric. This metric produces an invariant thermodynamic Riemannian curvature scalar R which, in fluid and spin systems, measures interatomic interactions. Specifically, |R| measures the size of organized fluctuating microscopic structures, and the sign of R indicates whether the interactions are effectively attractive or repulsive. R has also been calculated for black hole thermodynamics for which there is no consensus about any underlying microscopic structures. It is hoped that the physical interpretation of R in fluid and spin systems might offer insight into black hole microstructures. I give a brief review of results for R in black holes, including stability, the sign of R, R = 0, diverging |R|, and various claims of "inconsistencies" in thermodynamic metric geometry.

The analytic torsion of a cone over an odd dimensional manifold

We study the analytic torsion of a cone over an orientable odd dimensional compact connected Riemannian manifold W . We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We show that this last term coincides with the anomaly boundary term appearing in the Cheeger Müller theorem [3,2] for a manifold with boundary, according to Brüning and Ma (2006) [5]. We also prove Poincaré duality for the analytic torsion of a cone.

The decomposition of global conformal invariants IV: A proposition on local Riemannian invariants

This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern–Gauss–Bonnet integrand. The present paper lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these were used extensively in the first two papers in this series, see Alexakis (2007, 2009) [2,3]. These results may be of independent interest, applicable to related problems.

Single-Phase averaging for the Ablowitz-Ladik chain

The Bogolyubov-Whitham averaging method is applied to the Ablowitz-Ladik chain $$ \begin{gathered} - i\dot q_n - (1 - q_n r_n )(q_{n - 1} + q_{n + 1} ) + 2q_n = 0, \hfill \\ - i\dot r_n + (1 - q_n r_n )(r_{n - 1} + r_{n + 1} ) + 2r_n = 0 \hfill \\ \end{gathered} $$ in the single-phase case. We consider an averaged system and prove that the Hamiltonian property is preserved under averaging. The single-phase solutions are written in terms of elliptic functions and, in the “focusing” case, Riemannian invariants are obtained for modulation equations. The characteristic rates of the averaged system are stated in terms of complete elliptic integrals and the self-similar solutions of the systemare obtained. Results of the corresponding simulations are given.

Ricci curvature of Lagrangian submanifolds in complex space forms

As we showed in [5] and [6], the basic inequalities, involving Riemannian invariants of a Lagrangian submanifold immersed in a complex space form, can be improved using optimization methods. Also in [1] is showed that the improved Chen’s inequality from [5] is optimal. In this paper we find another proof for a Chen’s inequality, regarding the Ricci curvature [2] and we improve this inequality in the Lagrangian case. Mathematics subject classification (2010): 53C21, 53C24, 53C25, 49K35.

Renormalizing curvature integrals on Poincaré–Einstein manifolds

After analyzing renormalization schemes on a Poincaré–Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well known and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré–Einstein structure and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss–Bonnet theorem.

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.

Riemannian geometry on contact Lie groups

We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structures on Lie groups

Noncommutative Geometry and Lower Dimensional Volumes in Riemannian Geometry

In this paper we explain how to define “lower dimensional” volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential way noncommutative geometry and the analysis of Dirac operators on spin manifolds. However, the ultimate definitions of the lower dimensional volumes do not involve noncommutative geometry or spin structures at all.

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 .

Fixed points and determining sets for holomorphic self-maps of a hyperbolic manifold

We study fixed point sets for holomorphic automorphisms (and endomorphisms) on hyperbolic manifolds. The main object of our interest is to determine the number and configuration of fixed points that forces an automorphism (endomorphism) to be the identity. These questions have been examined in a number of papers for bounded domains in C. Here we extend these results to a finite dimensional hyperbolic manifold. In some important cases such extension is not obvious. A bounded domain can be equipped with an invariant Riemannian (Bergman) metric, and one can use differential geometry technics to obtain results. Such a metric is not always available on a general hyperbolic manifold. To overcome this obstacle we introduce locally a different invariant Hermitian metric.

Flag curvature of invariant Randers metrics on homogeneous manifolds

In this article we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds and by using a counter example we show that the formula which obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Randers metrics on the naturally reductive homogeneous manifolds $(G/H,g)$, where the Randers metric induced by the invariant Riemannian metric $g$ and an invariant vector field $\tilde{X}$ which is parallel with $g$.

On the bochner conformal curvature of Kähler-Norden manifolds

Using the one-to-one correspondence between Kähler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a Kähler-Norden manifold to be holomorphically recurrent.

On the curvature of Kähler–Norden manifolds

Using the one-to-one correspondence between Kähler–Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics are established. Especially, the holomorphic versions of the recurrence of the Riemann, Ricci, projective are defined and investigated. For four-dimensional Kähler–Norden manifolds, it is proved that they are of holomorphically recurrent curvature on the set where the holomorphic scalar curvature does not vanish. Furthermore, a four-dimensional Kähler–Norden manifold is (locally) conformally flat if and only if its holomorphic scalar curvature is constant pure imaginary. The present paper continues author’s investigations of Kähler–Norden manifolds from the papers [K. Słuka, On Kähler manifolds with Norden metrics, An. Ştiint. Univ. Al.I. Cuza IaşI Ser. Ia Mat. 47 (2001) 105–122; K. Słuka, Properties of the Weyl conformal curvature of Kähler–Norden manifolds, in: Proc. Colloq. Diff. Geom. on Steps in Differential Geometry, July 25–30, 2000, Debrecen, 2001, pp. 317–328].