Affine Immersions Research Articles

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.

1-Conformal geometry of quasi statistical manifolds

A quasi statistical manifold is a generalization of a statistical manifold. The notion of quasi statistical manifolds was introduced to formulate the geometry of non-conservative estimating functions in statistics. Later, it was showed that quasi statistical manifolds are induced from affine distributions in the same way as statistical manifolds are induced from affine immersions. Here, an affine distribution is a non-integrable version of an affine immersion, and it is useful in quantum information geometry. On the other hand, it is known that generalized conformal geometry is useful for the study of statistical manifolds from the viewpoint of affine differential geometry. In particular, 1-conformal geometry of statistical manifolds gives a relation with the notion of affine immersions. Although generalized conformal geometry of quasi statistical manifolds is also expected to be useful, the geometry has not been cleared yet. The aim of this paper is to formulate 1-conformal geometry of quasi statistical manifolds. We research a relation between 1-conformal geometry of quasi statistical manifolds and the notion of affine distributions. As the main result, we show the fundamental theorems for affine distributions. We also formulate a hypersurface theory of quasi statistical manifolds.

Twistor holomorphic affine surfaces and projective invariants

We study affine immersions with twistor lifts. Using a decomposition of a connection, we obtain several projective invariants for such affine immersions. In particular, affine immersions with holomorphic twistor lifts are considered. We can show the property that an affine immersion has holomorphic twistor lifts is invariant under projective transformations and characterize immersions with holomorphic twistor lifts by vanishing of some of projective invariants. In the case of compact affine surfaces with holomorphic twistor lifts, we see a quantization phenomenon for one of the projective invariants which we obtain. Moreover, we prove that a real analytic twistor holomorphic affine surface with the symmetric Ricci tensor with respect to both complex structures is totally geodesic or totally umbilic.

Parallel Affine Immersions $ {M^n}\to {{\mathbb{R}}^{n+2 }} $ with Flat Connection

We present a classification of parallel affine immersions f : $ {M^n}\to {{\mathbb{R}}^{n+2 }} $ Mn ! Rn + 2 with flat connection according to the rank of the Weingarten mapping.

Relative nullity distributions, an affine immersion from an almost product manifold and a para-pluriharmonic isometric immersion

We give a characterization for an affine immersion from an almost product manifold the almost product structure of which is adjoint or skew-adjoint with respect to its affine fundamental form by the relative nullity distribution.

Divisibility of multilinear mappings with applications to affine differential geometry

For real finite-dimensional vector spaces V, W we call a bilinear symmetric mapping h : V × V → W non-degenerate if the components of h with respect to a certain basis are linearly independent and non-degenerate. We say that a symmetric trilinear mapping C : V × V × V → W is divisible by h if there exists a linear form α such that C(v, v, v) = α(v)h(v, v) for every v ∈ V. In affine differential geometry of affine immersions h is the second fundamental form and C – the cubic form of the immersion. The immersion has pointwise planar normal sections if h(v, v) ∧ C(v, v, v) = 0 for every tangent vector v. We prove that it implies that C is divisible by h if h is non-degenerate and the codimension is greater than two. For immersions with Wiehe's or Sasaki's choice of transversal bundles divisibility of C by h implies vanishing of C.

A Characterization for a (2, 0)-Geodesic Affine Immersion

We give a characterization for a (2, 0)-geodesic affine immersion to an affine space by using its index of relative nullity. Especially, we prove a cylinder theorem for such a hypersurface. We also show a cylinder theorem for a (2, 0)-geodesic isometric immersion from a Kahler manifold and anti-Kahler manifold as a corollary.

Affine planar geodesic immersions with maximal codimension

This work gives a classification theorem for affine immersions with planar geodesics in the case where the codimension is maximal. Vrancken classified parallel affine immersions in this case and obtained, among others, generalized Veronese submanifolds. In this work it is shown that the immersions with planar geodesics are the same as the parallel ones in the considered case. A geometric interpretation of parallel immersions is also given: The affine immersions with pointwise planar normal sections (with respect to the equiaffine transversal bundle) are parallel. This result is verified for surfaces in ℝ4 and for immersions with the maximal codimension.

The Contact Number of an Affine Immersion and Its Upper Bounds

We generalize the contact number of a Euclidean submanifold to that of an affine immersion. We show that when the ambient space is projectively flat if the contact number of an affine immersion with codimension p is finite, then it is at most p + 1.

Relative Nullity Distributions and (1, 1)-Geodesic Affine Immersions

We characterize a (1, 1)-geodesic affine immersion from a complex manifold with complex affine connection to an affine space whose index of relative nullity is not less than 2m – 4 everywhere, where a complex manifold has dimension 2m.

Affine surfaces which admit several affine immersions in $\mathbb{R}^3$

Let $F: \Sigma \to \mathbb{R}^3$ be a Blaschke immersion of an affine surface $(\Sigma,\nabla)$ with a positive definite affine fundamental form such that $dim Im \, R = 1$ where $R$ is the curvature tensor. Suppose that there exists another immersion of the same surface with the same induced affine connection $\nabla$ which is not affine equivalent to the first one. Then we give explicitely $F$. Therefore all immersions which admit another immersion which is not affine equivalent to the original one are classified.

Blaschke structure for a special affine immersion

In this paper we determine a Blaschke structure for affine immersion of Euclidian and Hyperbolic type for plane equi-affine curve. In particular, we consider this structure in the case where the Ricci tensor of affine immersion is constant and give a necessary and sufficient condition for Ricci tensor to be constant.

THE GEOMETRY OF THE DIRICHLET MANIFOLD

In the present paper, we investigate the geometric structures of the Dirichlet manifold composed of the Dirichlet distribution. We show that the Dirichlet distribution is an exponential family distribution. We consider its dual structures and give its geometric metrics, and obtain the geometric structures of the lower dimension cases of the Dirichlet manifold. In particularly, the Beta distribution is a 2-dimensional Dirich-let distribution. Also, we construct an affine immersion of the Dirichlet manifold. At last, we give the e-flat hierarchical structures and the orthogonal foliations of the Dirichlet manifold. All these work will enrich the theoretical work of the Dirichlet distribution and will be great help for its further applications.

An existence theorem for $G$-structure preserving affine immersions

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

Neighbourhoods of independence and associated geometry in manifolds of bivariate Gaussian and Freund distributions

Abstract We provide explicit information geometric tubular neighbourhoods containing all bivariate distributions sufficiently close to the cases of independent Poisson or Gaussian processes. This is achieved via affine immersions of the 4-manifold of Freund bivariate distributions and of the 5-manifold of bivariate Gaussians. We provide also the α-geometry for both manifolds. The Central Limit Theorem makes our neighbourhoods of independence limiting cases for a wide range of bivariate distributions; the topological character of the results makes them stable under small perturbations, which is important for applications in models of stochastic processes.

A pseudo-inverse of the fundamental form and its application to affine immersions of non-degenerate surfaces

We choose a canonical transversal bundle of an affine immersion if a pseudo-inverse of the fundamental form exists. In particular, well-known canonical transversal bundles for a non-degenerate surface in ℝ4 are generalized to those for a non-degenerate surface in 4-dimensional manifold with torsion-free connection.

A semi-parallel fundamental form of maximal rank for a decomposition of a vector bundle with connection

We study a subbundle with semi-parallel fundamental form. In particular, if the rank of the fundamental form is maximal, we can obtain a certain equation which plays an essential role to classify parallel affine immersions into ℝn+12n(n+1)

Decompositions of para-complex vector bundles and para-complex affine immersions

In this work we study decompositions of para-complex and para-holomorphic vector-bundles endowed with a connection ∇ over a para-complex manifold. First we obtain results on the connections induced on the subbundles, their second fundamental forms and their curvature tensors. In particular we analyze para-holomorphic decompositions. Then we introduce the notion of para-complex affine immersions and apply the above results to obtain existence and uniqueness theorems for para-complex affine immersions. This is a generalization of the results obtained by Abe and Kurosu [AK] to para-complex geometry. Further we prove that any connection with vanishing (0, 2)-curvature, with respect to the grading defined by the para-complex structure, induces a unique para-holomorphic structure.

Pluriharmonic maps in affine differential geometryand $(1, 1)$-geodesic affine immersions

We define a pluriharmonic map from a complex manifold with a complex affine connection to a manifold with an affine connection and obtain some fundamental results which generalize those for a pluriharmonic map from a K\"{a}hler manifold to a Riemannian manifold. Especially, by using an associated family, we find a sufficient condition for the product of two $(1,1)$-geodesic affine immersions to an affine space to be a complex affine immersion from the manifold to the product of affine spaces with a certain complex structure.

Locally homogeneous affine connections on compact surfaces

Global properties of locally homogeneous and curvature homogeneous affine connections are studied. It is proved that the only locally homogeneous connections on surfaces of genus different from 1 are metric connections of constant curvature. There exist nonmetrizable nonlocally symmetric locally homogeneous affine connections on the torus of genus 1. It is proved that there is no global affine immersion of the torus endowed with a nonflat locally homogeneous connection into R 3 {\mathbf R} ^3 .