Invariant Einstein metrics on three-locally-symmetric spaces

Abstract

Abstract. In this paper, we classify three-locally-symmetric spaces for a connected, compact and simple Lie group. Furthermore, we give the classification of invariant Einstein metrics on these spaces.

The evolution of positively curved invariant Riemannian metrics on the Wallach spaces under the Ricci flow

“…The complete classification of generalized Wallach spaces is obtained recently (independently) in the papers [15,25]. For a fixed bi-invariant inner product ·, · on the Lie algebra g of the Lie group G, any G-invariant Riemannian metric g on G/H is determined by an Ad(H )-invariant inner product …”

New non-naturally reductive Einstein metrics on exceptional simple Lie groups

“…Let G/K be a reductive homogeneous space, where G is a semi-simple compact connected Lie group, K is a connected closed subgroup of G, g and k are the corresponding Lie algebras, respectively. If m, the tangent space of G/K at o = π(e), can be decomposed into three ad(k)-invariant irreducible summands pairwise orthogonal with respect to B as: In [12] and [22], the authors gave the complete classification of generalized Wallach spaces in the simple case. Here we use the notations in [12], then for p = 2, there are E 6 -III and E 8 -II, for p = 3, there are F 4 -II and E 8 -I and for p = 4 there is only E 7 -I.…”

“…According to the article [12], each kind of generalized Wallach spaces arising from simple Lie groups is associated with two commutative involutive automorphisms of g, the Lie algebra of G. With these two involutive automorphisms, we have two different corresponding decompositions of g, which are in fact irreducible symmetric pairs. According to these two irreducible symmetric pairs, we can get some linear equations for the non-zero coefficients in the expression of components of Ricci tensor with respect to the given metric.…”

“…With the help of computer, we get the Einstein metrics from the solutions of systems polynomial equations. We mainly deal with two kinds of generalized Wallach spaces, one of which is without centers in k and the other is with a 1-dimensional center in k. Along with the results in [11], we list all the number of non-naturally reductive left-invariant Einstein metrics on exceptional simple Lie groups G arising from generalized Wallach spaces with no center in k. In this table, we still use the [12] to represent the type of generalized Wallach space, N non−nn represents the number of non-naturally reductive Einstein metrics on G and p, q coincides with the indices in the decomposition g = k 1 ⊕ · · · ⊕ k p ⊕ m 1 ⊕ · · · ⊕ m q , in fact q = 3 for all types.…”

Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces

“…A complete classification of such spaces is given in [13,18]. They possess a number of interesting properties, and their geometry has been studied by several authors; see, e.g., [7,1,13,8]. There are several infinite families and 10 isolated examples, excluding products, constructed out of exceptional Lie groups.…”