Differential Geometric Invariants Research Articles

The Bézier curve and neural network model of the time-domain transient signals

The discussion is for the memory of Oleg Alexandrovich Tretyakov. In the discussion, one characteristic of the signal transfer of Professor Tretyakov's Evolutionary Approach to the Electromagnetics method is presented. Theoretically, the problem of the signal generated by a time-domain signal in a waveguide is addressed. The theoretic propagation is realized-exemplified through the actual TEC (TECU) map estimating by the Bézier curve and neural network. The striking aspect of the segmented prototype established with the Bézier approach is its adaptability. This mechanical curve, which does not need any preliminary preparation, is framed on differential geometric invariants. In the essay, a time-dependent complete set of magnetic waveguide modes is remembered. While the Dirichlet and Neumann eigenvalue problems determine the vector functions of the modes, the behavior of the time-evolving amplitudes is given by the Klein-Gordon equation. Examples of current data are discussed with the TEC map for the 2017 year. While the actual fluctuation of the interpolated CODE TEC atlas is illustrated, the mechanical Bézier curve family (untouched before) and the neural network introduce time-domain estimations to the reader. The parametric curve approach governs the Bézier model. The curve, which is C0 class segmented continuously, models on its way with new hourly components every twelve hours. The network model employs the solar wind parameters for the TEC atlas estimation. The reliability and consistency of the models are exhibited by the R correlation ratio, absolute error, and mean squared error. As a result, the R coefficient of the curve and network models vary around 91.2% and 98.8%, respectively. One can note the error of the network model falls to 1.1308 TECU. The outcomes are compatible with the former discussions.

Exotic cyclic cohomology classes and Lipschitz algebras

We study the noncommutative geometry of algebras of Lipschitz continuous and H\"older continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic $K$-theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi's relative sequence in $K$-theory.

Footvector Representation of Curves and Surfaces

This paper proposes a foot mapping-based representation of curves and surfaces which is a geometric generalization of signed distance functions. We present a first-order characterization of the footvector mapping in terms of the differential geometric invariants of the represented shape and quantify the dependence of the spatial partial derivatives of the footvector mapping with respect to the principal curvatures at the footpoint. The practical applicability of foot mapping representations is highlighted by several fast iterative methods to compute the exact footvector mapping of the offset surface of CSG trees. The set operations for footpoint mappings are higher-order functions that map a tuple of functions to a single function, which poses a challenge for GPU implementations. We propose a code generation framework to overcome this that transforms CSG trees to the GLSL shader code.

Tensor invariants of generalized Kenmotsu manifolds

In this paper, we study the properties of generalized Kenmotsu manifolds, consider the second-order differential geometric invariants of the Riemannian curvature tensor of generalized Kenmotsu manifolds (by the symmetry properties of the Riemannian geometry tensor). The concept of a tensor spectrum is introduced. Nine invariants are singled out and the geometric meaning of these invariants turning to zero are investigated. The identities characterizing the selected classes are singled out. Also, 9 classes of generalized Kenmotsu manifolds are distinguished, the local structure of 8 classes from the selected ones is obtained.

Local differential geometry of cuspidal edge and swallowtail

We investigate the local differential geometric invariants of cuspidal edge and swallowtail from the view point of singularity theory. We introduce finite type invariants of such singularities (see Remark 1.5 and Theorem 2.11) based on certain normal forms for cuspidal edge and swallowtail. Then we discuss several geometric aspects based on our normal form. We also present several asymptotic formulas concerning our invariants with respect to Gauss curvature and mean curvature.

Dualities of Differential Geometric Invariants on Cuspidal Edges on Flat Fronts in the Hyperbolic Space and the de Sitter Space

We compute the differential geometric invariants of cuspidal edges on flat surfaces in hyperbolic 3-space and in de Sitter space. Several dualities of invariants are pointed out.

Geometry of cuspidal edges with boundary

We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invariants.

Generalized Similar Frenet Curves

The paper is devoted to differential geometric invariants determining a Frenet curve in En up to a direct similarity. These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then, they expressed by focal curvatures of the curve. And then, we give the relationship between curvatures of evolute curve and shape curvatures. Morever, the geometric interpretations of these invariants are given.

Differential geometric invariants for time-reversal symmetric Bloch-bundles: The “Real” case

Topological quantum systems subjected to an even (resp. odd) time-reversal symmetry can be classified by looking at the related “Real” (resp. “Quaternionic”) Bloch-bundles. If from one side the topological classification of these time-reversal vector bundle theories has been completely described in De Nittis and Gomi [J. Geom. Phys. 86, 303–338 (2014)] for the “Real” case and in De Nittis and Gomi [Commun. Math. Phys. 339, 1–55 (2015)] for the “Quaternionic” case, from the other side it seems that a classification in terms of differential geometric invariants is still missing in the literature. With this article and its companion [G. De Nittis and K. Gomi (unpublished)] we want to cover this gap. More precisely, we extend in an equivariant way the theory of connections on principal bundles and vector bundles endowed with a time-reversal symmetry. In the “Real” case we generalize the Chern-Weil theory and we show that the assignment of a “Real” connection, along with the related differential Chern class and its holonomy, suffices for the classification of “Real” vector bundles in low dimensions.

Geometric Invariants of Cuspidal Edges

AbstractWe give a normal form of the cuspidal edge that uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges that determine them up to order three. We also clarify relations between these invariants.

Singularities of null developable of timelike curve that lies on nullcone in semi-Euclidean 3-space with index 2

In this paper, we investigate the null developables of timelike curves that lie on nullcone in 3-dimensional semi-Euclidean space with index 2. We classify the singularities of the null developables of timelike curves. The primary approach is based on the classical unfolding theory in singularity theory, which has been extensively applied in studying singularity problems in Euclidean and semi-Euclidean space. The study shows that the differential geometric invariants of timelike curves measured the order of the contact between a timelike curve and a conic CN contained in NCo⁎. Finally, an example is provided to explain our findings.

Singularities of Focal Surfaces of Null Cartan Curves in Minkowski 3‐Space

Singularities of the focal surfaces and the binormal indicatrix associated with a null Cartan curve will be investigated in Minkowski 3‐space. The relationships will be revealed between singularities of the above two subjects and differential geometric invariants of null Cartan curves; these invariants are deeply related to the order of contact of null Cartan curves with tangential planar bundle of lightcone. Finally, we give an example to illustrate our findings.

Null Darboux developable and pseudo-spherical Darboux image of null Cartan curve in Minkowski 3-space

Singularities of null Darboux developable, Gaussian surfaces and pseudo-spherical Darboux images associated with a null Cartan curve will be investigated in Minkowski 3-space. The relationships will be revealed between singularities of the above three subjects and differential geometric invariants of null Cartan curves, these invariants are deeply related to the order of contact of null Cartan curves with null helices.

Singularities of ruled null surfaces of the principal normal indicatrix to a null Cartan curve in de Sitter 3-space

In this Letter, we investigate the ruled null surfaces of the principal normal indicatrix of a null Cartan curve in de Sitter 3-space, an important vacuum solution to Einstein's equations of general relativity with cosmological terms. We classify the singularities of the ruled null surfaces and reveal the relationships between the singularities of the ruled null surfaces and the differential geometric invariants of null Cartan curves by applying the singularity theory. The primary approach is based on the classical unfolding theorem in singularity theory, which has been extensively applied in studying singularity problems in Euclidean space and Minkowski space. This study shows that singularities of the ruled null surfaces and differential geometric invariants of null Cartan curves are deeply related to the order of the contact between the binormal indicatrix of null Cartan curve and the pseudo-sphere contained in the nullcone.

SHIING-SHEN CHERN 1911–2004

Shiing-Shen Chern was born on October 26th, 1911 in Jia Xin, Chekiang Province, in China. His father practised law and worked for the government. At Fu Luen Middle School in Tsientsin he first showed his mathematical ability by doing all the exercises in classical English textbooks on algebra and trigonometry, and then went at the age of fifteen to Nankai University — a one-man Department run by Li-Fu Chiang, a student of Coolidge. As a result he studied a great deal of geometry, reading Coolidge, Salmon, Castelnuovo and Staude. He then became a postgraduate in 1930 at Tsinghua University in Beijing (or Peiping as it was then called) and came under the influence of Dan Sun, one of the few mathematicians in China writing research papers. During this period he became seriously interested in Sun's subject, projective differential geometry, and studied in detail the works of Wilhelm Blaschke. It was also at Tsinghua that he met his wife Shih-Ning, the daughter of a professor. After Blaschke paid a visit to Tsinghua in 1932 and lectured on differential-geometric invariants, Chern won a fellowship to study with him in Hamburg for two years. In 1936 he received his DSc there for work on the theory of webs. While in Hamburg, he also attended the lectures of Kähler on what is now called Cartan–Kähler theory, and the following year he spent in Paris studying with Cartan himself.

Differential-Geometric Invariants of the Hamiltonian Systems of pde’s

Differential-geometric invariants of Hamiltonian systems of partial differential equations are introduced on the basis of the Schouten (2,1)-tensor \(S^{jk}_i\). Algebraic identities for the corresponding Schouten, Nijenhuis and Haantjes tensors are found. The necessary conditions for existence of Hamiltonian structures for systems of hydrodynamic type are derived.

Differential geometry of G1 variable radius rolling ball blend surfaces

Variable radius rolling ball (VRRB) blend surfaces can be considered as envelopes of one parameter families of varying radius balls. These families are usually G 1 continuous but often only piecewise curvature continuous. In this paper these (spherical) VRRB surfaces will be analyzed on the basis of the theory of envelopes and discriminant sets. The differential geometric invariants of the VRRB surface are determined and the progressive and regressive points on the VRRB surface are characterized. The concept of geodesic blend surface is introduced and analyzed which avoids local selfintersections if both surfaces locally enclose the variable radius balls.

On the maslov index

AbstractIn this paper we give four definitions of Maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other. Moreover, relationships of several symplectic and differential geometric, analytic, and topological invariants (including triple Maslov indices, eta invariants, spectral flow and signatures of quadratic forms) to the Maslov index are developed and formulae relating them are given. The broad presentation is designed with a view to applications both in geometry and in analysis. © 1994 John Wiley & Sons, Inc.

Examples of constant mean curvature immersions of the 3-sphere into euclidean 4-space

Mean curvature is one of the simplest and most basic of local differential geometric invariants. Therefore, closed hypersurfaces of constant mean curvature in euclidean spaces of high dimension are basic objects of fundamental importance in global differential geometry. Before the examples of this paper, the only known example was the obvious one of the round sphere. Indeed, the theorems of H. Hopf (for immersion of S(2) into E(3)) and A. D. Alexandrov (for imbedded hypersurfaces of E(n)) have gone a long way toward characterizing the round sphere as the only example of a closed hypersurface of constant mean curvature with some added assumptions. Examples of this paper seem surprising and are constructed in the framework of equivariant differential geometry.

Some relations between differential geometric invariants and topological invariants of submanifolds

Let M be an n-dimensional manifold immersed in an m-dimensional euclidean space Em and let ∇ and ∇̃ be the covariant differentiations of M and Em, respectively. Let X and Y be two tangent vector fields on M. Then the second fundamental form h is given by(1.1) ∇̃XY = ∇XY + h(X,Y).