Source Manifold Research Articles

A DDVV Conjecture for Riemannian Maps

The Wintgen inequality is a significant result in the field of differential geometry, specifically related to the study of submanifolds in Riemannian manifolds. It was discovered by Pierre Wintgen. In the present work, we deal with the Riemannian maps between Riemannian manifolds that serve as a superb method for comparing the geometric structures of the source and target manifolds. This article is the first to explore a well-known conjecture, called DDVV inequality (a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken), for Riemannian maps, where we consider different space forms as target manifolds. There are numerous research problems related to such inequality in various ambient manifolds. These problems can all be explored within the general framework of Riemannian maps between various Riemannian manifolds equipped with notable geometric structures.

Tangent functor on microformal morphisms, and non-linear pullbacks for forms and cohomology

We show how the tangent functor extends from ordinary smooth maps to “microformal morphisms” (also called “thick morphisms”) of supermanifolds. Microformal morphisms generalize ordinary maps and correspond to formal canonical relations between the cotangent bundles specified by generating functions depending on position variables on the source manifold and momentum variables on the target manifold (as formal power expansions), regarded as part of the structure. Microformal morphisms act on functions by non-linear (in general) pullbacks. We obtain here non-linear pullbacks of (pseudo)differential forms and show that they respect the de Rham differentials as “non-linear chain maps” that can induce non-linear transformations of cohomology.

Deep Domain Adaptation Using Cascaded Learning Networks and Metric Learning

How can we bridge efficiently distribution difference between the source and target domains in an isomorphic latent feature space by using metric learning. This study introduces metric learning on manifolds which combine a cascaded learning network and a metric learning model to form a Unified Domain Adaptation Model. Our approach is based on formulating a transfer from the source to the target as a geometric mean metric learning problem on manifolds. The solution of symmetric positive definite covariance matrices not only reduces the statistical differences between the source and target domains, but also the underlying geometry of the source and target domains using diffusions on the underlying source and target manifolds. By retaining both the nonlinear structure of the Riemannian geometry of the open cone of symmetric positive definite matrices and cascaded learning networks, we improve the state-of-the-art results on the Amazon (A), Caltech256 (C), DSLR (D), Webcam (W) and VisDA benchmark datasets by knowledge transfer, while achieving comparable performances to competing methods on domain adaptation modeling.

Fold cobordisms and a Poincaré–Hopf-type theorem for the signature

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author's earlier works. Secondly we take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into R^n. Then we are looking for dependencies between these invariants and we study fold maps of 4k-dimensional manifolds into R^2. We construct special fold maps which are representatives of the fold cobordism classes and we also compute cobordism groups. We obtain a Poincare-Hopf type formula, which connects local data of the singularities of a fold map of an oriented 4k-dimensional manifold M to the signature of M. We also study the unoriented case analogously and prove a similar formula about the twisting of the normal bundle of the fold singular set.

4-Manifold topology, Donaldson–Witten theory, Floer homology and higher gauge theory methods in the BV-BFV formalism

We study the behavior of Donaldson’s invariants of 4-manifolds based on the moduli space of anti-self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang–Mills theory, known today as Donaldson–Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable [Formula: see text]-dimensional field theories (in the sense of Baez–Dolan and Lurie) in the perturbative setting, especially when [Formula: see text]. Additionally, we relate these constructions to Nekrasov’s partition function by treating an equivariant version of Donaldson–Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov–Witten and Donaldson–Thomas theory and recall their correspondence conjecture for general Calabi–Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena.

Stability of non-proper functions

The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney topology). We show that a Morse function is stable if it is end-trivial at any point in its discriminant, where end-triviality (which is also called local triviality at infinity) is a property concerning behavior of functions around the ends of the source manifolds. We further show that a Morse function is strongly stable if (and only if) it is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we give a sufficient condition for stability of Nash functions, and show that any Nash function becomes stable after a generic linear perturbation.

Conformality of rotationally symmetric maps

Let (M,g), (N,h) be Riemannian manifolds without boundary, and let f be a smooth map from M into N. We consider a covariant symmetric tensor Tf = f⁎h−1m‖df‖2g, where f⁎h denotes the pullback of the metric h by f, and m is the dimension of the manifold M. The tensor Tf vanishes if and only if the map f is a weakly conformal map. The norm ‖Tf‖ is a quantity which is a measure of the conformality of f at each point. In [4] the author introduced maps which are critical points of the functional Econf(f) = ∫M‖Tf‖2dvg in the case that M is compact. We call such maps C-stationary maps. In the case that M is non-compact, f is defined to be a C-stationary map if it is C-stationary on any compact subset of M.Any conformal map or more generally any weakly conformal one is a C-stationary map. However, there exists a C-stationary map which is not a weakly conformal one. (See Example in [3], p. 156.) In the case that the source manifold or the target one is the n-dimensional standard sphere (n≥5), any stable C-stationary map is a weakly conformal one. (See Theorem 1 and 2 in [3].)In this paper we are concerned with rotationally symmetric maps. We prove that any rotationally symmetric smooth map between 4-dimensional model spaces is a C-stationary map if and only if it is a conformal one.

Notes About a Harmonicity on the Tangent Bundle with Vertical Rescaled Metric

In this article, we present some results concerning the harmonicity on the tangent bundle equipped with the vertical rescaled metric. We establish necessary and sufficient conditions under which a vector field is harmonic with respect to the vertical rescaled metric and we construct some examples of harmonic vector fields. We also study the harmonicity of a vector field along with a map between Riemannian manifolds, the target manifold is equipped with a vertical rescaled metric on its tangent bundle. Next we also discuss the harmonicity of the composition of the projection map of the tangent bundle of a Riemannian manifold with a map from this manifold into another Riemannian manifold, the source manifold being whose tangent bundle is endowed with a vertical rescaled metric. Finally, we study the harmonicity of the tangent map also the harmonicity of the identity map of the tangent bundle.

CARLSON–GRIFFITHS THEORY FOR COMPLETE KÄHLER MANIFOLDS

AbstractWe investigate Carlson–Griffiths’ equidistribution theory of meormorphic mappings from a complete Kähler manifold into a complex projective algebraic manifold. By using a technique of Brownian motions developed by Atsuji, we obtain a second main theorem in Nevanlinna theory provided that the source manifold is of nonpositive sectional curvature. In particular, a defect relation follows if some growth condition is imposed.

Formal global perturbative quantization of the Rozansky–Witten model in the BV-BFV formalism

We describe a globalization construction for the Rozansky–Witten model in the BV-BFV formalism for a source manifold with and without boundary in the classical and quantum case. We define an AKSZ sigma model, which, upon globalization through notions of formal geometry extended appropriately to our case, is shown to reduce to the Rozansky–Witten model. The relations with other relevant constructions in the literature are discussed. Moreover, we split the model as a BF-like theory and we construct a perturbative quantization of the model in the quantum BV-BFV framework. In this context, we are able to prove the modified differential Quantum Master Equation and the flatness of the quantum Grothendieck BFV operator. Additionally, we provide a construction of the BFV boundary operator in some cases.

A Survey of Unsupervised Generative Models for Exploratory Data Analysis and Representation Learning

For more than a century, the methods for data representation and the exploration of the intrinsic structures of data have developed remarkably and consist of supervised and unsupervised methods. However, recent years have witnessed the flourishing of big data, where typical dataset dimensions are high and the data can come in messy, incomplete, unlabeled, or corrupted forms. Consequently, discovering the hidden structure buried inside such data becomes highly challenging. From this perspective, exploratory data analysis plays a substantial role in learning the hidden structures that encompass the significant features of the data in an ordered manner by extracting patterns and testing hypotheses to identify anomalies. Unsupervised generative learning models are a class of machine learning models characterized by their potential to reduce the dimensionality, discover the exploratory factors, and learn representations without any predefined labels; moreover, such models can generate the data from the reduced factors’ domain. The beginner researchers can find in this survey the recent unsupervised generative learning models for the purpose of data exploration and learning representations; specifically, this article covers three families of methods based on their usage in the era of big data: blind source separation, manifold learning, and neural networks, from shallow to deep architectures.

BV and BFV for the H-Twisted Poisson Sigma Model

We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form H. There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using an auxiliary connection $$\nabla $$ on the target manifold M, we obtain alternative, purely geometrical expressions without the use of superfields, which are new also for the ordinary PSM and promise adaptations to other Lie algebroid-based gauge theories: The BV functional, in particular, is the sum of the classical action, the Hamiltonian lift of the (only on-shell nilpotent) BRST differential, and a term quadratic in the antifields which is essentially the basic curvature and measures the compatibility of $$\nabla $$ with the Lie algebroid structure on $$T^*M$$ . We finally construct a $$\hbox {Diff}(M)$$ -equivariant isomorphism between the two BV formulations.

The size of some vanishing and critical sets

We prove that the vanishing sets of all top forms on a non-orientable manifold are at least 1-dimensional in the general case and at most $1$-codimen\-sional in the compact case. We apply these facts to show that the critical sets of some differentiable maps are at least 1-dimensional in the general case and at most 1-codimensional when the source manifold is compact.

Eells–Sampson Type Theorems for Subelliptic Harmonic Maps from sub-Riemannian Manifolds

In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we prove some Eells–Sampson type existence results for this flow when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub-Riemannian structure comes from a tense Riemannian foliation. Finally, some Hartman type results are also established for the flow.

J-holomorphic curves and Dirac-harmonic maps

Dirac-harmonic maps are critical points of a fermionic action functional, generalizing the Dirichlet energy for harmonic maps. We consider the case where the source manifold is a closed Riemann surface with the canonical Spinc-structure determined by the complex structure and the target space is a Kähler manifold. If the underlying map f is a J-holomorphic curve, we determine a space of spinors on the Riemann surface which form Dirac-harmonic maps together with f. For suitable complex structures on the target manifold the tangent bundle to the moduli space of J-holomorphic curves consists of Dirac-harmonic maps. We also discuss the relation to the A-model of topological string theory.

Two applications of the Schwarz lemma

The Schwarz lemmas are well-known characterizations for holomorphic maps and we exhibit two examples of their applications. For a sequence family of biholomorphisms $f_j$, it is useful to determine the location of $f_j(q)$ for a fixed point $q$ in source manifolds (see Proposition \ref{2.5}). With it, we extend the Fornaess-Stout's theorem of \citep{FS77} in monotone unions of balls to ellipsoids in Section \ref{sec2}. In Section \ref{sec3}, we discuss the curvature bounds of complete Kahler metric on $\rtimes$ domains defined in \citep{Liu004} with an idea of \citep{Ya76}.

Short time existence of the heat flow for Dirac-harmonic maps on closed manifolds

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtained short time existence, and the existence of a global weak solution was established by Jost, Liu, and Zhu. We prove short time existence of the heat flow for Dirac-harmonic maps on closed manifolds.

Solving variational problems and partial differential equations that map between manifolds via the closest point method

Maps from a source manifold M to a target manifold N appear in liquid crystals, color image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems and partial differential equations (PDEs) that map between manifolds is introduced within this paper. Our approach, the closest point method for manifold mapping, reduces the problem of solving a constrained PDE between manifolds M and N to the simpler problems of solving a PDE on M and projecting to the closest points on N. In our approach, an embedding PDE is formulated in the embedding space using closest point representations of M and N. This enables the use of standard Cartesian numerics for general manifolds that are open or closed, with or without orientation, and of any codimension. An algorithm is presented for the important example of harmonic maps and generalized to a broader class of PDEs, which includes p-harmonic maps. Improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods. Harmonic and p-harmonic maps are computed for a variety of numerical examples. In these examples, we denoise texture maps, diffuse random maps between general manifolds, and enhance color images.

Elimination of cusps in dimension 4 and its applications

We study a class of homotopies between maps from 4-manifolds to surfaces which we call cusp merges. These homotopies naturally appear in the uniqueness problems for certain pictorial descriptions of 4-manifolds derived from maps to the 2-sphere (for example, broken Lefschetz fibrations, wrinkled fibrations, or Morse 2-functions). Our main results provide a classification of cusp merge homotopies in terms of suitably framed curves in the source manifold, as well as a fairly explicit description of a parallel transport diffeomorphism associated to a cusp merge homotopy. The latter is the key ingredient in understanding how the aforementioned pictorial descriptions change under homotopies involving cusp merges. We apply our methods to the uniqueness problem of surface diagrams of 4-manifolds and describe algorithms to obtain surface diagrams for total spaces of (achiral) Lefschetz fibrations and 4-manifolds of the form M × S 1 , where M is a 3-manifold. Along the way we provide extensive background material about maps to surfaces and homotopies thereof and develop a theory of parallel transport that generalizes the use of gradient flows in Morse theory.

Views of Real Projective 3-Space by Stable Maps into the Plane

ABSTRACTSmooth stable maps into the plane enable us to get the graphical views of the source manifolds. In this article, we present two such maps of the real projective 3-space constructed from typical two projections of a tetrahedron to the plane. Tri- and quarto-sections of the real projective 3-space and Heegaard splittings of genus four and five can be obtained naturally using these maps. A question is posed on views of projective n-spaces of n ⩾ 4.