It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric constant holomorphic sectional curvature. In this paper we give full classification all possible Riemannian metrics on group. We prove each those negative scalar curvature, only one them being Einstein (up to isometry and scaling).

We consider the shape of balls for nilpotent Lie groups endowed with a left invariant Riemannian or sub-Riemannian metric. We prove that when the algebra is graded these balls are homeomorphic to the standard Euclidean ball. For two-step nilpotent groups we show that the intersections of the ball with the central cosets are star-shaped, and in special cases convex.

Gromoll and Meyer have represented a certain exotic 7-sphere Σ as a biquotient of the Lie group G = Sp(2). We show for a 2-parameter family of left invariant metrics on G that the induced metric on Σ has strictly positive sectional curvature at all points outside four subvarieties of codimension ≥ 1 which we describe explicitly.

We demonstrate that the left (and right) invariant Maurer-Cartan forms for any semi-simple Lie group enable one to construct solutions of the Yang-Mills equations on the group manifold equipped with the natural Cartan-Killing metric. For the unitary unimodular groups the Yang-Mills action integral is finite for such solutions. This is explicitly exhibited for the case of SU(3).

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

Let L ⊂ V = R be a maximally isotropic subspace. It is shown that any simply connected Lie group with a bi-invariant flat pseudo-Riemannian metric of signature (k, l) is 2-step nilpotent and is defined by an element η ∈ Λ3L ⊂ Λ3V . If η is of type (3, 0)+(0, 3) with respect to a skew-symmetric endomorphism J with J2 = ǫId, then the Lie group L(η) is endowed with a left-invariant nearly Kähler s...

We answer in the affirmative question posed by Conti and Rossi [7,8] on existence of nilpotent Lie algebras dimension 7 with an Einstein pseudo-metric nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ signature $(3, 4)$ group 7, such that is not Ricci-flat. show cannot be induced any closed $G_2^*$-structure group. Moreover, some results harmonic $G_2^...