Abstract:Pseudo-Riemannian geometry is, to a large extent, the study of the Levi-Civita connection, which is the unique torsion-free connection compatible with the metric structure. There are, however, other affine connections which arise in different contexts, such as conformal geometry, contact structures, Weyl structures, and almost Hermitian geometry. In this book, we reverse this point of view and instead associate an auxiliary pseudo-Riemannian structure of neutral signature to certain affine connections and use … Show more
“…Weyl geometry is a generalization of Riemannian geometry where the metric tensor and the covariant derivative g μν , ∇ μ , are generalized to g μν ,∇ μ , where∇ μ is not defined by the Levi-Civita connection of g μν , but by the affine connection˜ κ μν (g) with the property [62]∇…”
We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-æther theory defined in Weyl integrable geometry. The existence of exact and analytic solutions is investigated for the case of a spatially flat Friedmann–Lemaître–Robertson–Walker background space. We show that the theory admits cosmological solutions of special interests. In addition, we prove that the cosmological field equations admit the Lewis invariant as a second conservation law, which indicates the integrability of the field equations.
“…Weyl geometry is a generalization of Riemannian geometry where the metric tensor and the covariant derivative g μν , ∇ μ , are generalized to g μν ,∇ μ , where∇ μ is not defined by the Levi-Civita connection of g μν , but by the affine connection˜ κ μν (g) with the property [62]∇…”
We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-æther theory defined in Weyl integrable geometry. The existence of exact and analytic solutions is investigated for the case of a spatially flat Friedmann–Lemaître–Robertson–Walker background space. We show that the theory admits cosmological solutions of special interests. In addition, we prove that the cosmological field equations admit the Lewis invariant as a second conservation law, which indicates the integrability of the field equations.
“…Riemannian extensions relate the affine geometry of (Σ, D) with the pseudo-Riemannian geometry of (T * Σ, g D,Φ ) and, moreover, they are the underlying structure of many Walker manifolds. For instance, self-dual Walker manifolds as well as Ricci-flat Walker manifolds are Riemannian extensions up to some modifications (see [9,31] for details).…”
Section: Walker Structures and Riemannian Extensionsmentioning
Abstract. It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.
“…Proof. We follow the treatment in [14] to construct the germ of a suitable structure and refer to [5,6] for a different treatment. We will then use Theorem 2.7 to transplant this structure into M .…”
We say that a germ G of a geometric structure can be transplanted into a manifold M if there is a suitable geometric structure on M which agrees with G on a neighborhood of some point P of M . We show for a wide variety of geometric structures that this transplantation is always possible provided that M does in fact admit some such structure of this type. We use this result to show that a curvature identity which holds in the category of compact manifolds admitting such a structure holds for germs as well and we present examples illustrating this result. We also use this result to show geometrical realization problems which can be solved for germs of structures can in fact be solved in the compact setting as well. MSC 2010: 53B20 and 53B35.
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