Related papers: Integration of 1-forms and connections
Let $(G,H,\sigma)$ be a symmetric pair and $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ the canonical decomposition of the Lie algebra $\mathfrak{g}$ of $G$. We denote by ${\nabla}^0$ the canonical affine connection on the symmetric space…
We define the notion of affine rigidity of a hypergraph and prove a variety of fundamental results for this notion. First, we show that affine rigidity can be determined by the rank of a specific matrix which implies that affine rigidity is…
In this note we look at the freeness for complex affine hypersurfaces. If $X \subset \mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated projective hypersurface, obtained by taking the closure of $X$ in $\mathbb{P}^n$, then…
In this paper we prove that all manifolds with affine connection are globally projectively equivalent to some space with equiaffine connection (equiaffine manifold). These manifolds are characterised by a symmetric Ricci tensor.
A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally…
One of the most appealing results of metric-affine gauge theory of gravity is a close parallel between the Riemann curvature two-form and the Cartan torsion two-form: While the former is the field strength of the Lorentz-group connection…
We derive the interaction of fermions with a dynamical space-time based on the postulate that the description of physics should be independent of the reference frame, which means to require the form-invariance of the fermion action under…
The duality between a higher curvature $f(R)$ gravity model and a scalar-tensor theory helps to bring out the role of the additional degree of freedom originating from the higher derivative terms in the gravity action. Such a degree of…
We construct new explicit vacuum solutions of quadratic metric-affine gravity. The approach of metric-affine gravity in using an independent affine connection produces a theory with 10+64 unknowns, which implies admitting torsion and…
Partial connections are (singular) differential systems generalizing classical connections on principal bundles, yielding analogous decompositions for manifolds with nonfree group actions. Connection forms are interpreted as maps…
Pre-geodesics of an affine connection are the curves that are geodesics after a reparametrization (the analogous concept in K\"ahler geometry is known as J-planar curves). Similarly, dual-geodesics on a Riemannian manifold are curves along…
We consider the most general Quadratic Metric-Affine Gravity setup in the presence of generic matter sources with non-vanishing hypermomentum. The gravitational action consists of all $17$ quadratic invariants (both parity even and odd) in…
$f(Q)$ and $f(T)$ gravity are based on fundamentally different geometric frameworks, yet they exhibit many similar properties. This article provides a comprehensive summary and comparative analysis of the various theoretical branches of…
The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras.
In this article, we focus on symmetric teleparallel gravity, a modification of General Relativity where gravity is described by the non-metricity of an affine connection, whose curvature and torsion vanish. In these theories, the…
We prove that an affine cone $X$ admits a surjective morphism from an affine space if and only if $X$ is unirational.
We find possible cosmological models of the Polynomial Affine Gravity described by connections that are either compatible or not with a metric. When possible, we compare them with those of General Relativity. We show that the set of…
We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying…
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine…
We prove equivalence of certain axiom sets for affine buildings. Along the lines a purely combinatorial proof of the existence of a spherical building at infinity is given. As a corollary we obtain that ``being an affine building'' is…