A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous space. The naturally reductive case is also treated. This completes the geometric calculation of the isometry group of naturally reductive spaces. In addition, we prove that for normal homogeneous spaces the set of fixed points of the full isotropy is a torus. As an application of our results it follows that the holonomy group of a homogeneous fibration is contained in the group of (canonically) affine transformations of the fibers, in particular this holonomy group is a Lie group (this is a result of Guijarro and Walschap).
The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.
We classify tube domains in $\mathbb C^{n+1}$ ($n\ge 1$) with affinely homogeneous base of their boundary and a.) with positive definite Levi form and b.) with Lorentzian type Levi form and affine isotropy of dimension at least $\frac{(n-2)(n-3)}2$.
In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present paper, homogeneous geodesics in Finsler homogeneous spaces are studied using the affine method, which was developed in earlier papers by the author. The mentioned statement is proved correctly and it is further proved that any homogeneous Berwald space or homogeneous reversible …
We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature $(2,2)$ manifolds, to conformally Einstein manifolds and also to new Einstein manifolds through a warped product construction.
Apart from global topological problems an affine homogeneous space is locally described by its curvature, its torsion and a slightly less tangible object called its connection in a given base point. Using this description of the local geometry of an affine homogeneous space we construct an algebraic variety $\mathfrak{M}(\mathfrak{gl}\,V)$, which serves as a coarse moduli space for the local isometry classes of affine homogeneous spaces of dimension dim V. Moreover we associate a $\mathrm{Sym}V^*$-comodule to a point in $\mathfrak{M}(\mathfrak{gl}\,V\,)$ and use its Spencer cohomology in order…
Hildebrand classified all semi-homogeneous cones in $\mathbb{R}^3$ and computed their corresponding complete hyperbolic affine spheres. We compute isothermal parametrizations for Hildebrand's new examples. After giving their affine metrics and affine cubic forms, we construct the whole associated family for each of Hildebrand's examples. The generic member of these affine spheres is given by Weierstrass $\mathscr{P}, ~ζ~\text{and} ~σ$ functions. In general any regular convex cone in $\mathbb{R}^3$ has a natural associated $S^1$-family of such cones, which deserve further studies.
We give a new short self-contained proof of the result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] classifying the locally homogeneous torsion free affine surfaces and the extension to the case of surfaces with torsion due to Arias-Marco and Kowalski [T. Arias-Marco and O. Kowalski, Classification of locally homogeneous linear connections with arbitrary torsion on 2-dimensional manifolds, Monatsh. Math. 153 (2008), 1-18.]. Our approach rests on a direct analysis of the affine Killing equatio…
Let $\mathcal{M}$ be a Type $\mathcal{A}$ affine surface. We show that $\mathcal{M}$ is linearly strongly projectively flat. We use the quasi-Einstein equation together with the condition that $\mathcal{M}$ is strongly projectively flat to examine to examine the geodesic completeness of $\mathcal{M}$.
We examine the space of solutions to the affine quasi--Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.
We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi-Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connection in the plane.
We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $\mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $\mathcal{B}$ surfaces which are not Type $\mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.
We examine the topology of various spaces of locally homogeneous affine manifolds which arise from the classification result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] as orbits of the action of $GL(2,\mathbb{R})$ (Type $\mathcal{A}$) and the $ax+b$ group (Type $\mathcal{B}$). We determine the topology of the spaces of Type $\mathcal{A}$ models in relation to the rank of the Ricci tensor. We determine the topology of the spaces of Type $\mathcal{B}$ models which either are flat or where the…
We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies in a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborh…
The space of invariant affine connections on every $3$-Sasakian homogeneous manifold of dimension at least $7$ is described. In particular, the remarkable subspaces of invariant affine metric connections, and the subclass with skew-torsion, are also determined. To this aim, an explicit construction of all $3$-Sasakian homogeneous manifolds is exhibited. The unique $3$-Sasakian homogeneous manifolds which admit nontrivial Einstein with skew-torsion invariant affine connections are those of dimension $7$, that is, $\mathbb{S}^7=\mathrm{Sp} (2)/ \mathrm{Sp(1)}$, $\mathbb{R} P^7=\mathrm{Sp}(2)/ \m…
An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting of homogeneous affine surfaces.
In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. In the present paper this affine method is refined and adapted to the pseudo-Riemannian case. Using this method and elementary topology it is proved that any homogeneous Lorentzian manifold of even dimension admits a light-like homogeneous geodesic. The method is illustrated in d…
The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function. In particular, the nonzero level sets of a homogeneous polynomial are proper affine spheres if some power of it equals a nonzero multiple of its Hessian determinant. The relative invariants of real forms of regular irreducible prehomogeneous vector spaces yield many such polynomials which are moreover irreducible. For example, the nonzero level sets of t…
The nonzero level sets in $n$-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the $n$th power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood-Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the …
Many extensions of General Relativity are based on considering metric and affine structures as independent properties of spacetime. This leads to the possibility of introducing torsion as an independent degree of freedom. In this article we examine the effects of torsion on the affine Killing vectors of two-dimensional manifolds. We give a complete description of the Lie algebras of affine Killing vector fields on homogeneous surfaces. This can be used in the search of non-metrizable surfaces of interest.
We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let $\nabla$ be a unimodular real analytic affine connection on a real analytic compact connected surface $M$. If $\nabla$ is locally homogeneous on a nontrivial open set in $M$, we prove that $\nabla$ is locally homogeneous on all of $M$.
Consider a graphed holomorphic surface $u=F(x,y)$ in $\mathbb{C}^3_{x,y,u}$ under the action of the affine transformation group $A(3)$. In 1999, Eastwood and Ezhov obtained a list of homogeneous models by determining possible tangential vector fields. Inspired by Olver's recurrence formulas, we study the algebra of $A(3)$ differential invariants of surfaces. We obtain necessary conditions for homogeneity of algebraic nature. Solving these conditions, we organise homogeneous models in inequivalent branches.
We study the long time behavior of the volume preserving $p$-flow in $\mathbb{R}^{n+1}$ for $1\leq p<\frac{n+1}{n-1}$. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving $p$-flow converges sequentially to the unit ball in the $C^{\infty}$ topology, modulo the group of special linear transformations.
A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natural way. For the case of the odd dimensional spheres $\mathbb{S}^{2n+1}$ viewed as homogeneous spaces of the special unitary groups, the classical Nomizu's Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the sphe…
We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly projective invariant. Moreover the maximal dimension is shown to be achieved if and only if the manifold is strongly projectively flat.
In this paper, we introduce a new structure, namely, affine Szabó connection. We prove that, on $2$-dimensional affine manifolds, the affine Szabó structure is equivalent to one of the cyclic parallelism of the Ricci tensor. A characterization for locally homogeneous affine Szabó surface is obtained. Examples of two- and three-dimensional affine Szabó manifolds are also given.
We show that the category of affine bundles over a smooth manifold M is equivalent to the category of affine spaces modelled on projective finitely generated C^\infty(M)-modules. Using this equivalence of categories, we are able to give an alternate proof of the main result of [13], showing that the characterization of vector bundles by means of their Lie algebras of homogeneous differential operators also holds for vector bundles of rank 1 and over any base manifolds.
We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces admitting exactly one invariant connection.
In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds $N^n$ contained in hypersurfaces $M^{n+1}$ of the $(n+2)$-space. We give condition under which this affine focal set is a regular hypersurface and, for curves in $3$-space, we describe its stable singularities. For a given Darboux vector field $ξ$ of the immersion $N\subset M$, one can define the affine metric $g$ and the affine normal plane bundle $\mathcal{A}$. We prove that the $g$-Laplacian of the position vector belongs to $\mathcal{A}$ if and only if $ξ$ is parallel. For umbil…
In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Finsler spaces of positive flag curvature to an algebraic problem. The main result of this paper is a classification of positively curved normal homogeneous Finsler spaces. It turns out that a coset space $G/H$ admits a positively curved normal homogeneous Finsler metric if and only…
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.
Equivalences under the affine group ${\rm Aff} (\mathbb{R}^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. Especially, homogeneous models and algebras of differential invariants in various branches have been fully understood. Then what about higher dimensions? We consider hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ graphed as $\{ u = F(x_1, \dots, x_n) \}$ whose Hessian matrix $(F_{x_i x_j})$, a relative affine inv…
We classify holomorphic isometric actions on complex space forms all whose orbits are Lagrangian submanifolds, up to orbit equivalence. The only examples are Lagrangian affine subspace foliations of complex Euclidean spaces, and Lagrangian horocycle foliations of complex hyperbolic spaces.
We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitably rescaled solutions converge sequentially to the unit ball in the $\mathcal{C}^{\infty}$ topology modulo $SL(n+1)$.
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theorem is then proved in the generic singularities framework.
This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, under certain conditions, on the set of convex bodies in terms of the Hausdorff distance. Our proofs rely on the existence of the Orlicz-Petty bodies and the uniform boundedness of the Orlicz-Petty bodies of a convergent sequence of convex bodies. Similar results for the nonhomog…
