Geodesic completeness of the left-invariant metrics on ℝ H n

UDC 514 We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. We show that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case.

Classification of left invariant metrics on 4-dimensional solvable Lie groups

In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

The Ricci pinching functional on solvmanifolds

We study the natural functional $F=\frac {\operatorname {scal}^2}{|\operatorname {Ric}|^2}$ on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension $n$. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of $F$ restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group, and beyond the unimodular case, we obtain the same result for almost-abelian Lie groups. Many other aspects of the behavior of $F$ are clarified.

New non-naturally reductive Einstein metrics on SO(n)

We obtain new invariant Einstein metrics on the compact Lie groups [Formula: see text] ([Formula: see text]) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on [Formula: see text] and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

Recovering a Rotation Matrix From Three Direction Cosines

In this paper, we propose a new parameterization method to represent rotation matrices using the angles ϕ→ recovered from the three direction cosines that lie on the diagonal. The map from the possible configuration space of the new variable ϕ→ to the solid ball model in axis-angle coordinates is constructed. We also introduce a bi-invariant metric and two left-invariant metrics for measuring the distance in configuration space which could be the foundation for path planning in ϕ→ space. We further analyze the Jacobian matrix and singularities to better understand the manipulability.