In this paper, the reduction of biharmonic map equation in terms of the Maurer-Cartan form for all smooth map of an arbitrary compact Riemannian manifold into a compact Lie group (G, h) with bi-invariant Riemannian metric h is obtained. By this formula, all biharmonic curves into compaqct Lie groups are determined, and all the biharmonic maps of an open domain of R with the conformal metric of ...

Abstract The reduced Hamiltonian system on T∗(SU(3)/SU(2)) is derived from a Riemannian geodesic motion on the SU(3) group manifold parameterised by the generalised Euler angles and endowed with a bi-invariant metric. Our calculations show that the metric defined by the derived reduced Hamiltonian flow on the orbit space SU(3)/SU(2) ≃ S5 is not isometric or even geodesically equivalent to the s...

In the manuscript, we discuss the blur kernel measurement giving the ground truth kernel and propose an effective kernel similarity (KS) metric. In this section, we provide examples and more results to justify the proposed kernel similarity metric in Section 3 of the manuscript. A good metric for kernel similarity should be shift and range (i.e., the size of kernel) invariant. Figure 1 shows th...

Let X be a G-space such that the orbit space X/G is metrizable. Suppose a family of slices is given at each point of X. We study a construction which associates, under some conditions on the family of slices, with any metric on X/G an invariant metric on X. We show also that a family of slices with the required properties exists for any action of a countable group on a locally compact and local...

The investigation of manifolds with non-negative sectional curvature is one of the classical fields of study in global Riemannian geometry. While there are few known obstruction for a closed manifold to admit metrics of non-negative sectional curvature, there are relatively few known examples and general construction methods of such manifolds (see [Z] for a detailed survey). In this context, it...

It is well-known that the αG(M)-invariant introduced by Tian plays an important role in the study of the existence of Kähler-Einstein metrics on complex manifolds with positive first Chern class ([T1], [T2], [TY]). Based on the estimate of αG(M)-invariant, Tian in 1990 proved that any complex surface with c1(M) > 0 always admits a Kähler-Einstein metric except in two cases CP2#1CP2 and CP2#2CP2...

The global holomorphic invariant αG(M) introduced by Tian [6], Tian and Yau [5] is closely related to the existence of Kähler-Einstein metrics. In his solution of the Calabi conjecture, Yau [11] proved the existence of a KählerEinstein metric on compact Kähler manifolds with nonpositive first Chern class. Kähler-Einstein metrics do not always exist in the case when the first Chern class is posi...

The main goal is to classify 4-dimensional real Lie algebras gwhich admit a para-hypercomplex structure. This is a step toward the classification of Lie groups admitting the corresponding left-invariant structure and therefore possessing a neutral, left-invariant, anti-self-dual metric. Our study is related to the work of Barberis who classified real, 4-dimensional simply-connected Lie groups w...

There is a general method, applicable in many situations, whereby a pseudo–Riemannian metric, invariant under the action of some Lie group, can be deformed to obtain a new metric whose geodesics can be expressed in terms of the geodesics of the old metric and the action of the Lie group. This method applied to Euclidean space and the unit sphere produces new examples of complete Riemannian metr...