Abstract:
Ambrose and Singer characterized connected, simply-connected and complete homogeneous Riemannian manifolds as Riemannian manifolds admitting a metric connection such that its curvature and torsion are parallel. The aim of this paper is to extend Ambrose-Singer Theorem to the general framework of locally homogeneous pseudo-Riemannian manifolds. In addition we study under which conditions a locally homogeneous pseudo-Riemannian manifold can be recovered from the curvature and their covariant derivatives at some point up to finite order. The same problem is tackled in the presence of a geometric structure.

Abstract:
In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525] to remove a positivity constraint on a combination of the Ricci and Riemannian curvatures in a previous $L^{d/2}$-energy gap result due to Gerhardt (2010) and a previous $L^\infty$-energy gap result due to Bourguignon, Lawson, and Simons (1981, 1979), as well as an $L^2$-energy gap result due to Nakajima (1987) for a Yang-Mills connection over the sphere, $S^d$, but with an arbitrary Riemannian metric.

Abstract:
We extend an $L^2$ energy gap result due independently to Min-Oo and Parker (1982) for Yang-Mills connections on principal $G$-bundles, $P$, over closed, connected, four-dimensional, oriented, smooth manifolds, $X$, from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, including metrics which are generic and where the topologies of $P$ and $X$ obey certain mild conditions and the compact Lie group, $G$, is $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$.

Abstract:
A family of naturally reductive pseudo-Riemannian spaces is constructed out of the representations of Lie algebras with ad-invariant metrics. We exhibit peculiar examples, study their geometry and characterize the corresponding naturally reductive homogeneous structure.

Abstract:
The classification of 4-dimensional naturally reductive pseudo-Riemannian spaces is given. This classification comprises symmetric spaces, the product of 3-dimensional naturally reductive spaces with the real line and new families of indecomposable manifolds which are studied at the end of the article. The oscillator group is also analyzed from the point of view of this classification.

Abstract:
We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group $(N, < \,,\,>_N)$, such that $< \,,\,>_N$ is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-invariant one.

Abstract:
A method, due to \'Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to $\real^4$. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.

Abstract:
This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved here that the simply connected Lie group $N$ can be constructed starting from a real representation $(\pi,\vv)$ of a certain Lie algebra $\ggo$. We study the geometry of $(N, \la \,,\,\ra_N)$ and we find the corresponding naturally reductive homogeneous structure. On the other hand, related to the case of degenerate center we provide another family of naturally reductive spaces, both non compact and also compact examples.

Abstract:
We generalize our previous results (Theorem 1 and Corollary 2 in arXiv:1412.4114) and Theorem 1 in arXiv:1502.00668) on the existence of an $L^2$-energy gap for Yang-Mills connections over closed four-dimensional manifolds and energies near the ground state (occupied by flat, anti-self-dual, or self-dual connections) to the case of Yang-Mills connections with arbitrary energies. We prove that for any principal bundle with compact Lie structure group over a closed, four-dimensional, Riemannian manifold, the $L^2$ energies of Yang-Mills connections on a principal bundle form a discrete sequence without accumulation points. Our proof employs a version of our {\L}ojasiewicz-Simon gradient inequality for the Yang-Mills $L^2$-energy functional from our monograph arXiv:1409.1525 and extensions of our previous results on the bubble-tree compactification for the moduli space of anti-self-dual connections arXiv:1504.05741 to the moduli space of Yang-Mills connections with a uniform $L^2$ bound on their energies.

Abstract:
This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on $\Heis_3(\RR)$ by isometries and we finally study some geometrical features on these spaces.