Differentiable Manifold Research Articles

Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces

Inspired by the idea of Colding and Minicozzi (Ann Math 182:755–833, 2015), we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature.

Invariant Translators of the Heisenberg Group

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean translators: we mention in particular that we describe the analogous of the tilted grim reaper cylinders, of the bowl solution and of translating catenoids, but some of them are not convex in contrast with a recent result of Spruck and Xiao (Complete translating solitons to the mean curvature flow in {mathbb {R}}^3 with non-negative mean curvature, arXiv:1703.01003v2, 2017) in the Euclidean space. Moreover we also prove some non-existence results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group.

Semi-Simplicity of a Lie Algebra of Isometries

A spray S on the tangent bundle TM with a n dimensional differentiable manifold M defines an almost product structure Γ (Γ2 =I, I being the identity vector 1-form) and decomposes the TTM space into a direct sum of horizontal space (corresponding to the eigenvalue +1) and vertical space (for the eigenvalue -1). The Lie algebra of projectable vector fields whose Lie derivative vanishes the spray S is of dimension at most n2 +n. The elements of the algebra belonging to the horizontal nullity space of Nijenhuis tensor of Γ form a commutative ideal of . They are not the only ones for any spray S. If S is the canonical spray of a Riemannian manifold, the symplectic scalar 2-form Ω which is the generator of the spray S defines a Riemannian metric g upon the bundle vertical space of TM. The Lie algebra of infinitesimal isometries which is written contained in is of dimension at most . The commutative ideal of is also that of . The Lie algebra of dimension superior or equal to three is semi-simple if and only if the nullity horizontal space of the Γ Nijenhuis tensor is reduced to zero. In this case, Ag is identical to . Mathematics Subject Classification (2010) 53XX • 17B66 • 53C08 • 53B05ns.

Functional analysis and exterior calculus on mixed-dimensional geometries

We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d + 1-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.

Invariant submanifolds in golden Riemannian manifolds

In this paper, we study invariant submanifolds of a golden Riemannian manifold with the aid of induced structures on them by the golden structure of the ambient manifold. We demonstrate that any invariant submanifold in a locally decomposable golden Riemannian manifold leaves invariant the locally decomposability of the ambient manifold. We give a necessary and sufficient condition for any submanifold in a golden Riemannian manifold to be invariant. We obtain some necessary conditions for the totally geodesicity of invariant submanifolds. Moreover, we find some facts on invariant submanifolds. Finally, we present an example of an invariant submanifold.

Light cone and Weyl compatibility of conformal and projective structures

In the literature different concepts of compatibility between a projective structure {mathscr {P}} and a conformal structure {mathscr {C}} on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. In an often cited paper (Ehlers et al. in: O’Raifertaigh (ed) General Relativity, Papers in Honour of J.L. Synge, Clarendon Press, Oxford, 2012) Ehlers/Pirani/Schild introduce still another criterion which is natural from the physical point of view: every light like geodesics of {mathscr {C}} is a geodesics of {mathscr {P}}. Their claim that this type of compatibility is sufficient for introducing a Weylian metric has recently been questioned (Trautman in Gen Relativ Gravit 44:1581–1586, 2012); (Vladimir in Commun Math Phys 329:821–825, 2014); as reported by Scholz (in: A scalar field inducing a non-metrical contribution to gravitational acceleration and a compatible add-on to light deflection, 2019). Here it is proved that the conjecture of EPS is correct.

On Traces of Operators Associated with Actions of Compact Lie Groups

Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth manifold $M$, we study the operation, which takes each operator $D$ on the ambient manifold to a certain operator on the submanifold. The latter operator is called the trace of $D$. More precisely, we study traces of operators, associated with actions of compact Lie groups on $M$. We show that traces of such operators are localized at special submanifolds in $X$ and study the structure of the traces on these submanifolds.

Nonperturbative definition of the standard models

The Standard Models contain chiral fermions coupled to gauge theories. It has been a long-standing problem to give such gauged chiral fermion theories a quantum non-perturbative definition. By classification of quantum anomalies and symmetric invertible topological orders via a mathematical cobordism theorem for differentiable and triangulable manifolds, and the existence of symmetric gapped boundary for the trivial symmetric invertible topological orders, we propose that Spin(10) chiral fermion theories with Weyl fermions in 16-dimensional spinor representations can be defined on a 3+1D lattice, and subsequently dynamically gauged to be a Spin(10) chiral gauge theory. As a result, the Standard Models from the 16n-chiral fermion SO(10) Grand Unification can be defined non-perturbatively via a 3+1D local lattice model of bosons or qubits. Furthermore, we propose that Standard Models from the 15n-chiral fermion SU(5) Grand Unification can also be realized by a 3+1D local lattice model of fermions.

Mollifier smoothing of C0-Finsler structures

A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow [0,\infty)$ defined on the tangent bundle of a differentiable manifold $M$ such that its restriction to each tangent space is an asymmetric norm. We use the convolution of $F$ with the standard mollifier in order to construct a mollifier smoothing of $F$, which is a one parameter family of Finsler structures $F_\varepsilon$ (of class $\mathit{C}^\infty$ on $TM\backslash 0$) that converges uniformly to $F$ on compact subsets of $TM$. We prove that when $F$ is a Finsler structure, then the Chern connection, the Cartan connection, the Hashiguchi connection, the Berwald connection and the flag curvature of $F_\varepsilon$ converges uniformly on compact subsets to the corresponding objects of $F$. As an application of this mollifier smoothing, we study examples of two-dimensional piecewise smooth Riemannian manifolds with nonzero total curvature on a line segment. We also indicate how to extend this study to the correspondent piecewise smooth Finsler manifolds.

The value of the work done by an isotropic vector force field along an isotropic curve

In the present paper we consider a 3-dimensional differentiable manifold M equipped with a Riemannian metric g and an endomorphism Q, whose third power is the identity and Q acts as an isometry on g. Both structures g and Q determine an associated metric f on (M,g,Q). The metric f is necessary indefinite and it defines isotropic vectors in the tangent space TpM at an arbitrary point p on M.The physical forces are represented by vector fields. We investigate physical forces whose vectors are in TpM on (M,g,Q). Moreover, these vectors are isotropic and they act along isotropic curves. We study the physical work done by such forces.

The Grassmannian of affine subspaces

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold, an algebraic variety, a vector bundle, a classifying space, among many more structures; furthermore; it affords an analogue of Schubert calculus and its (co)homology and homotopy groups may be readily determined. On the other hand, like the Euclidean space, the affine Grassmannian serves as a concrete computational platform on which various distances, metrics, probability densities may be explicitly defined and computed via numerical linear algebra. Moreover, many standard problems in machine learning and statistics --- linear regression, errors-in-variables regression, principal components analysis, support vector machines, or more generally any problem that seeks linear relations among variables that either best represent them or separate them into components --- may be naturally formulated as problems on the affine Grassmannian.

Metrics for network attack and defense effectiveness based on differential manifolds

Metrics for network attack and defense effectiveness based on differential manifolds

Geodesic Nets: Some Examples and Open Problems

Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere. In the first part of this paper, we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. Many of these open questions are about geodesic nets with edges of multiplicity one on the Euclidean plane. Our main focus is on relationships between the number of boundary vertices, the number of inner (or balanced) vertices, and some basic geometric characteristic of geodesic nets (such as the length or the imbalances at boundary vertices). The second part contains a new construction providing a partial answer for one of these questions: We describe an infinite family of geodesic nets with edges of multiplicity one on the Euclidean plane with a constant number (namely, 14) of boundary vertices and arbitrarily many inner (or balanced) vertices of degree The fact that all edges of the constructed geodesic nets have multiplicity one is not proven but strongly supported by numerical evidence obtained from experimentation.

A Note on the Periodic Structure of Transversal Maps on the Torus and Products of Spheres

Let X be a compact differentiable manifold. A $$C^1$$ map $$f:X\rightarrow X$$ is called transversal if for all positive integers m, the graph of $$f^m$$ intersects transversally the diagonal of $$X\times X$$ at (x, x) for any x fixed point of $$f^m$$. In the present article, we describe the periodic structure of transversal maps on the n-dimensional torus. In particular, we give conditions on the eigenvalues of the induced linear map on the first homology, in order that all sufficiently large odd numbers are periods of the map. We present similar results for transversal maps on products of spheres of the same dimension. Later we generalize these results for transversal self-maps on rational exterior spaces of rank n.

Densities of Currents on Non-Kähler Manifolds

AbstractWe give a natural generalization of the Dinh–Sibony notion of density currents in the setting where the ambient manifold is not necessarily Kähler. As an application, we show that the algebraic entropy of meromorphic self-maps of compact complex surfaces is a finite bi-meromorphic invariant.

Automatic detection of feather defects using Lie group and fuzzy Fisher criterion for shuttlecock production

Feathers are the main raw material of shuttlecock. The quality of feather is a key factor in shuttlecock production and detection of feather defects is a challenging problem. In this paper, an automatic detection for feather defects was proposed that relies on fuzzy, Lie Group and machine learning theory. First, the feather images were enhanced using fuzzy set. Then, a covariance matrix was constructed as features of the defect region and Riemannian metric was assigned on a Riemannian manifold with a Lie group structure. The mean covariance matrix with nonsymmetric structure was proved to have a Lie group structure. To establish the automatic detection of feather defects, this study introduced Riemann mean and membership degree into the machine learning method of Fisher linear discriminant to implement feature recognition. Experiments were performed to demonstrate the feasibility of features of defect region as a differential manifold.

Foliations by Spacelike Hypersurfaces on Lorentz Manifolds

In this work, we study the geometric properties of spacelike foliations by hypersurfaces on a Lorentz manifold. We find an equation that relates the foliation with the ambient manifold and applies it to investigate conditions for the leaves being totally umbilical or geodesic. Using the Maximum principle with the mentioned equation we obtain an obstruction for the existence of totally geodesic spacelike foliations in a spacetime with positive Ricci curvature on the direction N.

Kodaira dimension of almost Kähler manifolds and curvature of the canonical connection

The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kahler manifolds, providing an explicit computation for a family of almost Kahler threefolds on the differentiable manifold underlying a Nakamura manifold. We concentrate also on the link between Kodaira dimension and the curvature of the canonical connection of an almost Kahler manifold and show that in the previous example (and in another one obtained from a Kodaira surface) the Ricci curvature of the almost Kahler metric vanishes for all the members of the family.

Mean curvature of hypersurfaces in Killing submersions with bounded shadow

AbstractGiven a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: The ambient manifold admits a Killing submersion with unit‐length Killing vector field. The projection of the image of the immersion is bounded in the base manifold. The hypersurface is stochastically complete, or the immersion is proper.

Submanifolds of almost poly-Norden Riemannian manifolds

Our aim in the present paper is to initiate the study of submanifolds in an almost poly-Norden Riemannian manifold, which is a new type of manifold first introduced by Sahin [17]. We give fundamental properties of submanifolds equipped with induced structures provided by almost poly-Norden Riemannian structures and find some conditions for such submanifolds to be totally geodesics. We introduce some subclasses of submanifolds in almost poly-Norden Riemannian manifolds such as invariant and antiinvariant submanifolds. We investigate conditions for a hypersurface of almost poly-Norden Riemannian manifolds to be invariant and totally geodesic, respectively, by using the components of the structure induced by the almost poly-Norden Riemannian structure of the ambient manifold. We also obtain some characterizations for totally umbilical hypersurfaces and give some examples of invariant and noninvariant hypersurfaces.