Classical Differential Geometry Research Articles

Classifications of THA-surfaces in I^3

In classical differential geometry, the problem of obtaining Gaussian and mean curvatures of a surface is one of the most important problems. A surface M2 in I3 is a THA-surface of first type if it can be parameterized by r(s, t) = (s, t, Af(s + at)g(t) + B(f(s + at) + g(t))). A surface M2 in I3 is a THA- surface of second type if it can be parameterized by r(s, t) = (s, Af(s + at)g(t) + B(f(s + at) + g(t)), t), where A and B are non-zero real numbers [16, 17, 18]. In this paper, we classify two types THA-surfaces in the 3-dimensional isotropic space I3 and study THA-surfaces with zero curvature in I3.

Differentialgeometrie im Grossen

Over the past several decades, classical differential geometry has undergone a remarkable expansion, helped by the integration of tools and insights from neighboring fields like partial differential equations, complex analysis, and geometric topology. In keeping with the spirit of previous gatherings, this meeting aimed to bridge the gaps between researchers working in seemingly disparate subfields of differential geometry, illuminating the connections that unite them. Amongst other things, this meeting was centered around the theme of scalar curvature, which has recently emerged as a fundamental element across various fields, including differential geometry, metric geometry, topology, and complex geometry. This shared topic presented an ideal opportunity for scholars from these distinct areas to convene, discuss their individual progress, and foster a vibrant exchange of ideas.

Osculating paraboloid

Osculating paraboloid of second order have been studied in classical differential geometry. In this article we generalize this concept to osculating paraboloids of higher order. This yields a visualization of the local properties of a given surface which depend on the derivatives of higher order.

Polynomial Guidance for Impact-Time Control Against Maneuvering Targets

To address the issue of intercepting maneuvering targets at a specific time, a polynomial guidance method for impact-time control is proposed in this paper. Based on the relative virtual framework and the classical differential geometry curve theory, such method is divided into two parts: 1) the design of the relative trajectory-length-control (RTLC) guidance law against virtual stationary targets, and 2) the design of the prediction algorithms based on the guidance law or its characteristics. The former realizes RTLC, and the latter establishes the relationship between the desired relative trajectory length-to-go and the desired time-to-go, thus implementing impact-time control. Furthermore, based on the analytical properties of the guidance law in the relative arc-length domain, its performance, characteristics, and allowable impact time are analyzed. Finally, the effectiveness of the proposed guidance method and the validity of the theoretical findings are verified by numerical simulations results.

Differential geometric guidance law design for varying-speed missile

A unified differential geometric guidance law (DGGL) design framework for varying-speed missiles against stationary targets is proposed in this paper. The varying-speed missile guidance model is established in the arc-length domain with the help of the classical differential geometry curve theory. It is strictly proven that DGGL is a natural and intuitive choice for varying-speed missiles when intercepting stationary targets since the influence of missile speed is directly eliminated in the guidance model and also during the guidance law design process. Hence, the guidance performance of DGGL will remain the same, no matter how the missile speed changes. After that, the error dynamics approach, which is widely used in the guidance law design field for constant-speed missiles in the time domain, is extended to the arc-length domain for DGGL design. Besides, the nonlinearity of the guidance model is fully considered when designing DGGL, hence the guidance errors caused by linearizations and small-angle assumptions are avoided. Then, taking the representative optimal error dynamics (OED) as an example, the corresponding OED-based DGGLs for zero-effort-miss-control, impact-angle-control, and flying-range-control are respectively derived. Finally, the derived illustrative guidance laws are simulated and compared with their time-domain counterparts to verify the effectiveness and superiority of the proposed unified DGGL design framework.

Minimum-Effort Waypoint-Following Differential Geometric Guidance Law Design for Endo-Atmospheric Flight Vehicles

To improve the autonomous flight capability of endo-atmospheric flight vehicles, such as cruise missiles, drones, and other small, low-cost unmanned aerial vehicles (UAVs), a novel minimum-effort waypoint-following differential geometric guidance law (MEWFDGGL) is proposed in this paper. Using the classical differential geometry curve theory, the optimal guidance problem of endo-atmospheric flight vehicles is transformed into an optimal space curve design problem, where the guidance command is the curvature. On the one hand, the change in speed of the flight vehicle is decoupled from the guidance problem. In this way, the widely adopted constant speed hypothesis in the process of designing the guidance law is eliminated, and, hence, the performance of the proposed MEWFDGGL is not influenced by the varying speed of the flight vehicle. On the other hand, considering the onboard computational burden, a suboptimal form of the MEWFDGGL is proposed to solve the problem, where both the complexity and the computational burden of the guidance law dramatically increase as the number of waypoints increases. The theoretical analysis demonstrates that both the original MEWFDGGL and its suboptimal form can be applied to general waypoint-following tasks with an arbitrary number of waypoints. Finally, the superiority and effectiveness of the proposed MEWFDGGL are verified by a numerical simulation and flight experiments.

Тензор кривизны n-поверхности и ее сферического образа в En+k

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k > 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

Differentialgeometrie im Grossen

The field of classical differential geometry has expanded enormously over the last several decades, helped by the development of tools from neighboring fields such as partial differential equations, complex analysis and geometric topology. In the spirit of the previous meetings in the series, this meeting will bring together researchers from apparently separate subfields of differential geometry, but whose work is linked by common themes. In particular, this meeting will emphasize intrinsic geometric questions motivated by the classification and rigidity of global geometric structures and the interaction of curvature with the underlying geometry and topology.

The fundamental theorem for singular surfaces with limiting tangent planes

AbstractIn this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3‐space known as frontals. Also, we characterize in a simple way these singular surfaces and its fundamental forms with local properties in the differential of its parametrization and decompositions in the matrices associated to the fundamental forms. In particular, we introduce new types of curvatures that can be used to characterize wave fronts. The only restriction on the parametrizations that is assumed in several occasions is that the singular set has empty interior.

A note about the non-commutative three-sphere with torsion and non-metricity

The Levi-Civita connection usually adopted on the non-commutative three-sphere is generalized to a connection possessing torsion and a non-metricity of the Weyl kind. This leads to geometrical properties that are qualitatively different from the ordinary ones and, in particular, provides an example of non-commutative “teleparallelism”. The comparison is made with the analogous case in classical (i.e. commutative) differential geometry. A general framework is developed, from which the unique connection having a given metric, torsion and non-metricity can be determined.

One‐parameter families of Legendre curves and plane line congruences

AbstractFamilies of curves in the Euclidean plane naturally contain singular curves, where the frame of classical differential geometry does not work well. We introduce the notions of one‐parameter family of Legendre curves in the Euclidean plane, congruent equivalence and curvature. Especially, a one‐parameter family of Legendre curves can contain singular curves, and is determined by the curvature up to congruence. We also give properties of one‐parameter families of Legendre curves. As applications, we give a relation between one‐parameter families of Legendre curves and Legendre surfaces. Moreover, we study plane line congruences (one‐parameter families of lines in plane) in terms of the curvatures as one‐parameter families of Legendre curves.

Deformation Modeling and Simulation of a Novel Bionic Software Robotics Gripping Terminal Driven by Negative Pressure Based on Classical Differential Algorithm.

A general pneumatic soft gripper is proposed in this paper. Combined with the torque balance theory, the mathematical theoretical model of bending deformation of soft gripper is established based on Yeoh constitutive model and classical differential geometry. Assuming that the pressure in each inner cavity is evenly distributed, the input gas is in an ideal state, which is approximately treated as an isothermal condition, and all orifices experience blocked flow. In addition, compared with the mechanical work of gas, the energy related to gas flow and heat transfer is negligible. The nonlinear mechanical properties of silicone rubber are studied. It is regarded as isotropic and incompressible material, which is characterized by strain energy per unit volume. The material constant coefficients C10 and C20 are determined through the uniaxial tensile test, and the software gripper is simulated on the ABAQUS platform. The bending deformation models of grippers with three different force-bearing cavity structures are analyzed and compared, and the software clamping structure with the bending deformation most in line with the application conditions is selected. The limit input air pressure of the gripper and the situation of enveloping the clamping target object are analyzed. Through the bending deformation experiment, the maximum deformation angle is 72.4°. The relative error between the simulation analysis data and the prediction results of the mathematical model is no more than 3.5%, which verifies the effectiveness of the simulation and the correctness of the mathematical theoretical model of bending deformation. The soft manipulator proposed in this paper has good adaptability to grasping objects of different shapes and sizes. The minimum diameter of the target object that can be clamped is 0.1 mm. It can clamp the object weighing up to 1 kg. It has compact size, light weight, high ductility, and flexibility.

Geometry of m-Hessian Equations

In the process of developing the modern theory of fully nonlinear, second-order partial differential equations, new geometric characteristics of surfaces naturally appeared. The implementation of these characteristics in terms of the classical differential geometry leads to significant technical difficulties. This paper provides a review of the necessary methodological reform and demonstrates a new differential geometric techniques by an example of constructing boundary barriers for m-Hessian equations.

Variational formulas for curves of fixed degree

We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [7, 21] who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.

Coordinate-free compatibility conditions for deformations of material surfaces

The study of material surfaces uses notions from classical differential geometry, such as the covariant gradient, the mean and Gaussian curvatures, and the Peterson–Mainardi–Codazzi and Gauss equations. These notions are traditionally introduced relative to local surface coordinates and involve Christoffel symbols. We proceed instead without recourse to coordinates using direct notation. After developing the formula for the covariant gradient relative to a surface metric, we derive versions of the Peterson–Mainardi–Codazzi and Gauss equations and Gauss’ Theorema Egregium relevant to a deformed material surface. We then apply our framework to kinematically constrained material surfaces. For material surfaces that can sustain only deformations that preserve either angles or lengths, we obtain explicit representations for the covariant gradient relative to the surface metric in terms of the surface gradient. We show also that a deformation of a material surface that preserves angles and areas must be length preserving and vice versa. Finally, we present an alternative derivation of the Peterson–Mainardi–Codazzi and Gauss equations for a deformed material surface subject to the provision that the surface metric derives from the metric for the ambient Euclidean space within which the surface is embedded. An Appendix involving coordinates is included to ease comparisons between our approach to covariant differentiation and associated derivations of the Peterson–Mainardi–Codazzi and Gauss equations and standard coordinate-based approaches.

On the classification of framed rectifying curves in Euclidean space

There are many studies on regular rectifying curves in classical differential geometry, and important results have been obtained. However, these studies are limited for a smooth curve with singular points. To examine such curves and surfaces, the concept of framed curve, which is the general form of regular and Legendre curves, is used. Framed curves are defined as curves that have a moving frame with singular points in Euclidean space. We investigate framed rectifying curves via the dilation of framed curves on S2 in . Moreover, the result of dilation of framed curves is the framed rectifying curve or not. We give some classifications for the dilation of framed curves. Finally, we give some related examples with their figures.

An encounter of classical differential geometry with dynamical systems in the realm of structural stability of principal curvature configurations

An encounter of classical differential geometry with dynamical systems in the realm of structural stability of principal curvature configurations

Functions with isotropic sections

We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if $n\geq 3$, $g:\mathbb{S}^{n-1}\to\mathbb{R}$ is an even bounded measurable function, $U$ is an open subset of $\mathbb{S}^{n-1}$ and the restriction (section) of $f$ onto any great sphere perpendicular to $U$ is isotropic, then ${\cal C}(g)|_U=c+\langle a,\cdot\rangle$ and ${\cal R}(g)|_U=c'$, for some fixed constants $c,c'\in\mathbb{R}$ and for some fixed vector $a\in \mathbb{R}^n$. Here, ${\cal C}(g)$ denotes the cosine transform and ${\cal R}(g)$ denotes the Funk transform of $g$. However, we show that $g$ does not need to be equal to a constant almost everywhere in $U^\perp:=\bigcup_{u\in U}(\mathbb{S}^{n-1}\cap u^\perp)$. For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.

Vector Bundles and Differential Bundles in the Category of Smooth Manifolds

A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced differential bundles in 2017 as an algebraic alternative to vector bundles in an arbitrary tangent category. In this paper, we prove that differential bundles in the category of smooth manifolds are precisely vector bundles. In particular, this means that we can give a characterisation of vector bundles that exhibits them as models of a tangent categorical essentially algebraic theory.

Parabolic regularity in geometric variational analysis

The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometry and geometric measure theory. The established results of second-order variational analysis and generalized differentiation, being married to the developed calculus of parabolic regularity, allow us to obtain novel applications to both qualitative and quantitative/numerical aspects of constrained optimization including second-order optimality conditions, augmented Lagrangians, etc. under weak constraint qualifications.