Theorem In Differential Geometry Research Articles

A discrete four vertex theorem for hyperbolic polygons

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a simple smooth closed curve in the plane has at least four extreme values. This theorem admits a natural discretization to Euclidean polygons due to O. Musin. In this article we adapt the techniques of Musin and prove a discrete four vertex theorem for convex hyperbolic polygons.

Some geometrical and topological properties of DNNs' decision boundaries

Geometry and topology of decision regions are closely related with classification performance and robustness against adversarial attacks. In this paper, we use differential geometry to theoretically explore the geometrical and topological properties of decision regions produced by deep neural networks (DNNs). The goal is to obtain some geometrical and topological properties of decision boundaries for given DNN models, and provide some principled guidance to design and regularization of DNNs. First, we present the curvatures of decision boundaries in terms of network parameters, and give sufficient conditions on network parameters for producing flat or developable decision boundaries. Based on the Gauss-Bonnet-Chern theorem in differential geometry, we then propose a method to compute the Euler characteristics of compact decision boundaries, and verify it with experiments.

The Effects of Finite Distance on the Gravitational Deflection Angle of Light

In order to clarify the effects of the finite distance from a lens object to a light source and a receiver, the gravitational deflection of light has been recently reexamined by using the Gauss–Bonnet (GB) theorem in differential geometry (Ishihara et al. 2016). The purpose of the present paper is to give a short review of a series of works initiated by the above paper. First, we provide the definition of the gravitational deflection angle of light for the finite-distance source and receiver in a static, spherically symmetric and asymptotically flat spacetime. We discuss the geometrical invariance of the definition by using the GB theorem. The present definition is used to discuss finite-distance effects on the light deflection in Schwarzschild spacetime for both the cases of weak deflection and strong deflection. Next, we extend the definition to stationary and axisymmetric spacetimes. We compute finite-distance effects on the deflection angle of light for Kerr black holes and rotating Teo wormholes. Our results are consistent with the previous works if we take the infinite-distance limit. We briefly mention also the finite-distance effects on the light deflection by Sagittarius A * .

Placement suitability criteria of composite tape for mould surface in automated tape placement

Automated tape placement is an important automated process used for fabrication of large composite structures in aeronautical industry. The carbon fiber composite parts realized with this process tend to replace the aluminum parts produced by high-speed machining. It is difficult to determine the appropriate width of the composite tape in automated tape placement. Wrinkling will appear in the tape if it does not suit for the mould surface. Thus, this paper deals with establishing placement suitability criteria of the composite tape for the mould surface. With the assumptions for ideal mapping and by applying some principles and theorems of differential geometry, the centerline trajectory of the composite tape is identified to follow the geodesic. The placement suitability of the composite tape is examined on three different types of non-developable mould surfaces and four criteria are derived. The developed criteria have been used to test the deposition process over several mould surfaces and the appropriate width for each mould surface is obtained by referring to these criteria.

A Serre–Swan theorem for bundles of bounded geometry

The Serre–Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. In this context it states that the category of Hilbert bundles of bounded geometry is equivalent to the category of operator ⁎-modules over the operator ⁎-algebra of continuously differentiable functions which vanish at infinity. Operator ⁎-modules are generalizations of Hilbert C⁎-modules where the category of C⁎-algebras has been replaced by a more flexible category of involutive algebras of bounded operators: The operator ⁎-algebras. Operator ⁎-modules play an important role in the study of the unbounded Kasparov product.

When action is not least for orbits in general relativity

In Newtonian mechanics, the action for a true trajectory between two spacetime events A and B is a minimum if the final event B occurs before the kinetic focus of the initial event A; otherwise, the action is a saddle point. We give a simplified proof of an analogous theorem in differential geometry for worldlines in a curved spacetime. We locate the kinetic focus for orbits in a Schwarzschild field in the lowest-order post-Newtonian approximation and show that the kinetic focus is shifted beyond its Newtonian value of one angular cycle by a fractional amount of order O(v2/c2).

Digital topological method for computing genus and the Betti numbers

This paper concerns with computation of topological invariants such as genus and the Betti numbers. We design a linear time algorithm that determines such invariants for digital spaces in 3D. These computations could have applications in medical imaging as they can be used to identify patterns in 3D image. Our method is based on cubical images with direct adjacency, also called ( 6 , 26 ) -connectivity images in discrete geometry. There are only six types of local surface points in such a digital surface. Two mathematical ingredients are used. First, we use the Gauss–Bonnett Theorem in differential geometry to determine the genus of 2-dimensional digital surfaces. This is done by counting the contribution for each of the six types of local surface points. The new formula derived in this paper that calculates genus is g = 1 + ( | M 5 | + 2 ⋅ | M 6 | − | M 3 | ) / 8 where M i indicates the set of surface-points each of which has i adjacent points on the surface. Second, we apply the Alexander duality to express the homology groups of a 3D manifold in the usual 3D space in terms of the homology groups of its boundary surface. While our result is stated for digital spaces, the same idea can be applied to simplicial complexes in 3D or more general cell complexes.

Energy integral in fracture mechanics (J‐integral) and Gauss‐Bonnet Theorem

AbstractThe J‐integral (a path‐independent energy integral) formalism is the standard method of analyzing nonlinear fracture mechanics. It is shown that the energy density of deformation fields in terms of the homotopy operator corresponds to the J‐integral for dislocation‐disclination fields and gives the force on dislocation‐disclination fields as a physical interpretation. The continuum theory of defects gives a natural framework for understanding the topological aspects of the J‐integral. This geometric interpretation gives that the J‐integral is an alternative expression of the well‐known theorem in differential geometry, i.e., the Gauss‐Bonnet theorem (with genus = 1). The geometrical expression of the J‐integral shows that the Eshelby's energy‐momentum (the physical quantity of the material space) is closely related to the Einstein 3‐form (the geometric objects of the material space).

Symmetrical Fundamental Tensors, Differential Operators, and Integral Theorems in Differential Geometry

To make the geometrical basis for soft matters with curved surfaces such as biomembranes as simple as possible, a symmetrical analytical system was developed in conventional differential geometry. The conventional second fundamental tensor is replaced by the so-called conjugate fundamental tensor. Because the differential properties of the conjugate fundamental tensor and the first fundamental tensor are symmetrical, the symmetrical analytical system including the symmetrical differential operators, symmetrical differential characteristics, and symmetrical integral theorems for tensor fields defined on curved surfaces can be constructed. From the symmetrical analytical system, the symmetrical integral theorems for mean curvature and Gauss curvature, with which the symmetrical Minkowski integral formulas are easily deduced just as special cases, can be derived. The applications of this symmetrical analytical system to biology not only display its simplicity and beauty, but also show its powers in depicting the symmetrical patterns of networks of biomembrane nanotubes. All these symmetrical patterns in soft matters should be just the reasonable and natural results of the symmetrical analytical system.

Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem

By appealing to the Poincare-Hopf Theorem on topological invariants, we introduce a global classification scheme for homogeneous, convex bodies based on the number and type of their equilibria. We show that beyond trivially empty classes all other classes are non-empty in the case of three-dimensional bodies; in particular we prove the existence of a body with just one stable and one unstable equilibrium. In the case of two-dimensional bodies the situation is radically different: the class with one stable and one unstable equilibrium is empty ( Domokos, Papadopoulos, Ruina, J. Elasticity 36 [1994], 59-66 ). We also show that the latter result is equivalent to the classical Four-Vertex Theorem in differential geometry. We illustrate the introduced equivalence classes by various types of dice and statistical experimental results concerning pebbles on the seacoast.

Derivation of the Evans Wave Equation from the Lagrangian and Action: Origin of the Planck Constant in General Relativity

The Evans wave equation is derived from the appropriate Lagrangian and action, identifying the origin of the Planck constant ħ in general relativity. The classical Fermat principle of least time, and the classical Hamilton principle of least action, are expressed in terms of a tetrad multiplied by a phase factor exp(iS/ħ), where S is the action in general relativity. Wave (or quantum) mechanics emerges from these classical principles of general relativity for all matter and radiation fields, giving a unified theory of quantum mechanics based on differential geometry and general relativity. The phase factor exp(iS/ħ) is an eigenfunction of the Evans wave equation and is the origin in general relativity and geometry of topological phase effects in physics, including the Aharonov-Bohm class of effects, the Berry phase, the Sagnac effect, related interferometric effects, and all physical optical effects through the Evans spin field B(3) and the Stokes theorem in differential geometry. The Planck constant ħ is thus identified as the least amount possible of action or angular momentum or spin in the universe. This is also the origin of the fundamental Evans spin field B(3), which is always observed in any physical optical effect. It originates in torsion, spin and the second (or spin) Casimir invariant of the Einstein group. Mass originates in the first Casimir invariant of the Einstein group. These two invariants define any particle.

An equality for the curvature function of a simple and closed curve on the plane

We prove an equality for the curvature function of a simple and closed curve on the plane. This equality leads to another proof of the four‐vertex theorem in differential geometry. While examining the regularity assumption on the curve for the equality, we make comments on the relation between the boundary regularity of a Riemann mapping and two important subjects, the Schauder theory and the strong maximum principle, for elliptic partial differential equations of second order. We take a note on the curvature function itself in recognizing people′s handwriting by a calculating device, as an afterthought on the equality.

The balance between existence and nonexistence theorems in differential geometry

We discuss the delicate balance between existence and nonexistence theorems in differential geometry. Studying their interplay yields some information about $ p $-harmonic maps, $ p $-SSU manifolds, geometric $ k_p $-connected manifolds, minimal hypersurfaces and Gauss maps, and manifolds admitting essential positive supersolutions of certain nonlinear PDE. As an application of the theory developed, we obtain a topological theorem for minimal submanifolds in complete manifolds with nonpositive sectional curvature.

Generic Phenomenological Theory of Vitrification

In the framework of the thermodynamics of irreversible processes and using a graphical approach based on the simple theorems of differential geometry, a generic phenomenological treatment of the glass transition is developed. A thermodynamic derivation of the basic kinetic condition for vitrification—the Frenkel–Kobeko–Reiner equation—is given in a new generalized form. The temperature course of the thermodynamic functions and the kinetic characteristics of vitrifying systems are constructed using one of the general principles of irreversible thermodynamics. The results thus obtained underline the substantial differences and formal similarities between vitrification, considered as a diffuse kinetic transition and thermodynamic phase transformations. A general reconsideration of the axiomatics of the thermodynamics and kinetics and kinetics of glass transition and relaxation is attempted.

Embedding of superstring backgrounds in Einstein gravity

A theorem of differential geometry is employed to locally embed a wide class of superstring backgrounds that admit a covariantly constant null Killing vector field in eleven-dimensional, Ricci-flat spaces. Included in this class are exact type IIB superstring backgrounds with non-trivial Ramond-Ramond fields and a class of supersymmetric string waves. The embedding spaces represent exact solutions to eleven-dimensional, vacuum Einstein gravity. A solution of eleven-dimensional supergravity is also embedded in a twelve--dimensional, Ricci-flat space.

A Method for Proving Theorems in Differential Geometry and Mechanics

A Method for Proving Theorems in Differential Geometry and Mechanics

Automated reasoning in differential geometry and mechanics using the characteristic set method

We clarify the formulation problem of mechanical theorem proving in differential geometry and mechanics and propose two formulations. We present complete methods of mechanical theorem proving for the two formulations. We also introduce predicates and a language to translate geometry statements into differential polynomial equations. A program based on our methods has proved more than 100 nontrivial theorems in differential geometry and elementary mechanics including various classification theorems for space curves, Bertrand's Theorem, Newton's gravitational laws, etc.

Generalization of physical component boundary fitted co‐ordinate (PCBFC) method for the analysis of free‐surface flow

AbstractThis paper presents a systematic and theoretically consistent approach for the analysis of free‐surface flow, making use of a number of established ideas such as physical component, boundary‐fitted co‐ordinate (BFC) and Lagrangian front tracking. The approach extends, theoretically as well as numerically, the use of physical component to general non‐orthogonal moving grids and provides a numerically stable BFC method with little labour of free‐surface positioning, grid generation and grid renewal. The approach conserves mass even at the free surface and allows time step of the order of the Coulant number. The main body of the present paper starts with the definition of analytical space and Riemannian geometry intrinsic to the physical component by applying to it the theorems of differential geometry and manifold theory. Then the governing equations of flow and free surface for the physical component are defined in the general 3D form with the notation of the new Riemannian geometry. Numerical procedures and the fully discrete equations are also presented for the benefit of potential users. Finally, several 2D examples demonstrate the basic performance of the present method by showing the computability of complex free‐surface motion.

Some new domains with complete Kähler metrics of negative curvature

We add to the known examples of complete Kahler manifolds with negative sectional curvature by showing that the following three classes of domains in euclidean spaces also belong: perturbations of ellipsoidal domains in ℂn, intersections of complex-ellipsoidal domains in ℂ2, and intersections of fractional linear transforms of the unit ball in ℂ2. In the process, we prove the following theorem in differential geometry: in the intersection of two complex-ellipsoidal domains in ℂ2, the sum of the Bergman metrics is a Kahler metric with negative curvature operator.

A remark on the Bochner technique in differential geometry

It is observed that by pushing the standard arguments one step further, almost all the theorems in differential geometry proved with the help of Bochner’s technique can be sharpened.