Affine Manifolds Research Articles

The vortex equation on affine manifolds

Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, \phi), consisting of a flat vector bundle E over M and a flat nonzero section \phi of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional reduction techniques for holomorphic pairs on K\ahler manifolds to the situation of flat pairs on affine manifolds.

The Riemann extensions with cyclic parallel Ricci tensor

The property of being a D'Atri space (i.e., a Riemannian manifold with volume‐preserving geodesic symmetries) is equivalent, in the real analytic case, to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition L3 is said to be an L3‐space. This definition extends easily to the affine case. Here we investigate the torsion‐free affine manifolds and their Riemann extensions as concerns heredity of the condition L3. We also incorporate a short survey of the previous results in this direction, including also the topic of D'Atri spaces.

Approximate Yang-Mills-Higgs metrics on flat Higgs bundles over an affine manifold

Given a flat Higgs vector bundle (E ,∇ , φ) over a compact connected special affine manifold, we first construct a natural filtration of E, compatible with both ∇ and φ, such that the successive quotients are polystable flat Higgs vector bundles. This is done by combining the Harder–Narasimhan filtration and the socle filtration that we construct. Using this filtration, we construct a smooth Hermitian metric h on E and a smooth 1–parameter family {At}t∈R of C∞ automorphisms of E with the following property. Let ∇ and φ be the flat connection and flat Higgs field respectively on E constructed from ∇ and φ using the automorphism At. If θ denotes the extended connection form on E associated to the triple h, ∇ and φ, then as t −→ +∞, the connection form θ converges in the C∞ Frechet topology to the extended connection form θ on E given by the affine Yang–Mills–Higgs metrics on the polystable quotients of the successive terms in the above mentioned filtration. In particular, as t −→ +∞, the curvature of θ converges in the C∞ Frechet topology to the curvature of θ.

CURVATURE OF MULTIPLY WARPED PRODUCTS WITH AN AFFINE CONNECTION

In this paper, we study the Einstein multiply warped prod- ucts with a semi-symmetric non-metric connection and the multiply warp- ed products with a semi-symmetric non-metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric non-metric connection and generalized Kasner spacetimes with a semi-symmetric non-metric connection and find some new examples of Einstein affine manifolds and affine manifo lds with constant scalar curvature. We also consider the multiply warped products with an affine connection with a zero torsion.

FIXED POINTS AND KERNEL OF THE PROJECTIVE HOLONOMY OF AN AFFINE MANIFOLD

It is an interesting problem to study fixed points of an element in the holonomy group of an affine manifold. We compute the limit of a sequence of projective transformations and verify relations between fixed points and kernels.

Affine projective Osserman structures

By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the deformed Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank-1-symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank-1 Riemannian symmetric space with the deformed Riemannian extension metric. We construct other examples of affine projective Osserman manifolds where the Ricci tensor is not symmetric and thus the connection in question is not the Levi-Civita connection of any metric. If the dimension is odd, we use methods of algebraic topology to show the Jacobi operator of an affine projective Osserman manifold has only one non-zero eigenvalue and that eigenvalue is real.

Universal Lagrangian bundles

The obstruction to construct a Lagrangian bundle over a fixed integral affine manifold was constructed by Dazord and Delzant (J Differ Geom 26:223–251, 1987) and shown to be given by ‘twisted’ cup products in Sepe (Differ Geom Appl 29(6): 787–800, 2011). This paper uses the topology of universal Lagrangian bundles, which classify Lagrangian bundles topologically [cf. Sepe in J Geom Phys 60:341–351, 2010], to reinterpret this obstruction as the vanishing of a differential on the second page of a Leray-Serre spectral sequence. Using this interpretation, it is shown that the obstruction of Dazord and Delzant depends on an important cohomological invariant of the integral affine structure on the base space, called the radiance obstruction, which was introduced by Goldman and Hirsch (Trans Am Math Soc 286(2):629–649, 1984). Some examples, related to non-degenerate singularities of completely integrable Hamiltonian systems, are discussed.

Flat bundles on affine manifolds

We review some recent results in the theory of affine manifolds and bundles on them. Donaldson–Uhlenbeck–Yau type correspondences for flat vector bundles and principal bundles are shown. We also consider flat Higgs bundles and flat pairs on affine manifolds. A bijective correspondence between polystable flat Higgs bundles and solutions of the Yang–Mills–Higgs equation in the context of affine manifolds is shown. Also shown, in the context of affine manifolds, is a bijective correspondence between polystable flat pairs and solutions of the vortex equation.

On metrizability of locally homogeneous affine 2-dimensional manifolds

In [19] we proved a theorem which shows how to find, under particular assumptions guaranteeing metrizability (among others, recurrency of the curvature is necessary), all (at least local) pseudo-Riemannian metrics compatible with a given torsion-less linear connection without flat points on a two-dimensional affine manifold. The result has the form of an implication only; if there are flat points, or if curvature is not recurrent, we have no good answer in general, which can be also demonstrated by examples. Note that in higher dimension, the problem is not easy to solve. Here we try to apply this apparatus to the two main types (A and B from [9], [1]) of torsion-less locally homogeneous connections defined in open domains of 2-manifolds. We prove that in dimension two a symmetric linear connection with constant Christoffels is metrizable if and only if it is locally flat. On the other hand, in the class of connections of type B there are even non-flat metrizable connections.

Almost Osserman structures on natural Riemann extensions

We study natural Einstein Riemann extensions of torsion-free affine manifolds (M,∇). Such a Riemann extension of n-dimensional (M,∇) is always a pseudo-Riemannian manifold of signature (n,n). It is well known that, if the base manifold (M,∇) is a torsion-free affine two-manifold with skew-symmetric Ricci tensor, or a flat affine space, we obtain a (globally) Osserman structure on the cotangent bundle T⁎M over (M,∇). If the new base manifold is an arbitrary direct product of the simple affine manifolds described above, we found that the resulting structures on T⁎M are not Osserman but only “almost Osserman”, in the sense that the Jacobi operator has to be restricted from the whole set of space-like unit vectors (or time-like unit vectors, respectively) to a complement of a subset of measure zero. We also find that the characteristic polynomial of the (restricted) Jacobi operator in the cotangent bundle depends only on the full dimension n of the base manifold, and it is the same as for the flat affine space.

Geodesics, Parallel Transport & One-Parameter Subgroups for Diffeomorphic Image Registration

Computational anatomy aims at developing models to understand the anatomical variability of organs and tissues. A widely used and validated instrument for comparing the anatomy in medical images is non-linear diffeomorphic registration which is based on a rich mathematical background. For instance, the large deformation diffeomorphic metric mapping (LDDMM) framework defines a Riemannian setting by providing a right invariant metric on the tangent spaces, and solves the registration problem by computing geodesics parametrized by time-varying velocity fields. A simpler alternative based on stationary velocity fields (SVF) has been proposed, using the one-parameter subgroups from Lie groups theory. In spite of its better computational efficiency, the geometrical setting of the SVF is more vague, especially regarding the relationship between one-parameter subgroups and geodesics. In this work, we detail the properties of finite dimensional Lie groups that highlight the geometric foundations of one-parameter subgroups. We show that one can define a proper underlying geometric structure (an affine manifold) based on the canonical Cartan connections, for which one-parameter subgroups and their translations are geodesics. This geometric structure is perfectly compatible with all the group operations (left, right composition and inversion), contrarily to left- (or right-) invariant Riemannian metrics. Moreover, we derive closed-form expressions for the parallel transport. Then, we investigate the generalization of such properties to infinite dimensional Lie groups. We suggest that some of the theoretical objections might actually be ruled out by the practical implementation of both the LDDMM and the SVF frameworks for image registration. This leads us to a more practical study comparing the parameterization (initial velocity field) of metric and Cartan geodesics in the specific optimization context of longitudinal and inter-subject image registration.Our experimental results suggests that stationarity is a good approximation for longitudinal deformations, while metric geodesics notably differ from stationary ones for inter-subject registration, which involves much larger and non-physical deformations. Then, we turn to the practical comparison of five parallel transport techniques along one-parameter subgroups. Our results point out the fundamental role played by the numerical implementation, which may hide the theoretical differences between the different schemes. Interestingly, even if the parallel transport generally depends on the path used, an experiment comparing the Cartan parallel transport along the one-parameter subgroup and the LDDMM (metric) geodesics from inter-subject registration suggests that our parallel transport methods are not so sensitive to the path.

Fitting cylinders to data

The problem of fitting cylinders to data arises in science and industry. This article proves the existence of generalized cylinders–Cartesian products of generalized spheres and affine manifolds–fitted to data by using many criteria, including generalized least-squares, weighted median, and midrange regressions.

Invariance of the naturally lifted metrics on linear frame bundles over affine manifolds

The natural lifts from affine manifolds to linear frame bundles have been classified by the second author in 1988. In this paper we prove the invariance Theorem for these natural lifts. But, at the beginning, we insert a short survey about some �prominent� natural lifts studied earlier by the present authors.

HERMITIAN–EINSTEIN CONNECTIONS ON PRINCIPAL BUNDLES OVER FLAT AFFINE MANIFOLDS

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle EG over M admits a Hermitian–Einstein structure if and only if EG is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.

Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

Abstract We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.

ROBUST EYE TRACKING IN VIDEO SEQUENCE

In this paper, we present a novel method for eye tracking, in detail describing the eye contour and the visible iris center. Combining the IVT (Incremental Visual Tracking) tracker, the proposed online affine manifold model, in which the sequentially learning shape and texture are modeled in the first stage and noniterative recovering estimation in the second stage, tracks the eye contour in video sequences. After that, an adaptive black round mask is generated to match the visible iris center. Experimental results of eye tracking indicate that our tracker works well in the PC or domestic camera captured image streams with considerable head and eyeball rotation.

Affine manifolds with diagonal holonomy

As first defined by Smillie, an affine manifold with diagonal holonomy is a manifold equipped with an atlas such that the changes of charts are restrictions of elements of the subgroup of Aff (\({\mathbb{R}^n}\)) formed by diagonal matrices. Refining Smillie’s theorem, Benoist proved that if a compact manifold M is split into manifolds with corners corresponding to complete simplicial fans of a fixed frame by its hypersurfaces with normal crossing, then the product of M and a torus of suitable dimension is a finite cover of an affine manifold with diagonal holonomy, and conversely. Motivated by the result of Benoist, we introduce a “Benoist manifold” and a natural definition of complexity for them. In particular, we study some properties of “Benoist manifolds”.

Triangulations and Volume Form on Moduli Spaces of Flat Surfaces

In this paper, we study the moduli spaces of flat surfaces with cone singularities verifying the following property: there exists a union of disjoint geodesic tree on the surface such that the complement is a translation surface. Those spaces can be viewed as deformations of the moduli spaces of translation surfaces in the space of flat surfaces. We prove that such spaces are quotients of flat complex affine manifolds by a group acting properly discontinuously, and preserving a parallel volume form. Translation surfaces can be considered as a special case of flat surfaces with erasing forest, in this case, it turns out that our volume form coincides with the usual volume form (which are defined via the period mapping) up to a multiplicative constant. We also prove similar results for the moduli space of flat metric structures on the n-punctured sphere with prescribed cone angles up to homothety. When all the angles are smaller than 2π, it is known (cf. [T]) that this moduli space is a complex hyperbolic orbifold. In this particular case, we prove that our volume form induces a volume form which is equal to the complex hyperbolic volume form up to a multiplicative constant.

The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called “Geodesic Lemma”, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds.

Mirror symmetry via logarithmic degeneration data, II

This paper continues the authors’ program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yau spaces. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B B . We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B B are described in terms of certain cohomology groups of sheaves on B B , as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B B , and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry and gives an explicit description of the monodromy of a smoothing.