Zero mean curvature submanifolds as generalizations of rotational surfaces in Minkowski space

The Poincare group is the group of inhomogeneous Lorentz transformations, namely Lorentz transformations followed by space-time translations. In order to study this group, we have to understand first the group of Lorentz transformations, the group of translations, and how these two groups are combined to form the Poincare group.

Reflections, spinors, and projections on a Minkowski space underlie Dirac's equation

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

The ability to manufacture complex design geometries via Additive Manufacturing (AM) has led to a rapid growth in advancing the design methods, fabrication, and application of Triply Periodic Minimal Surface (TPMS) lattices with minimal surface topologies. Due to its zero-mean curvature, TPMS lattices can be additively manufactured without any sacrificial support structures and offer both design and manufacturing engineers, unprecedented control over the local physical properties (surface area, relative density, etc.) and local mechanical properties (flexural strength, Young’s modulus, etc.). TPMS lattices are of high interest for a wide range of applications such as biomedical implants, energy absorption, and surface fluidic applications such as heat exchangers, and energy storage. Recent advancements in functionally graded TPMS lattice design by varying local lattice geometry has shown to result in different mechanical performance. However, there have been limited studies in understanding the functional grading of AM process conditions (e.g., Laser-Powder Bed Fusion in this study) and lattice sheet thickness to better map the design-processing conditions-properties. The goal of this study is to achieve similar mechanical properties in TPMS sheet lattices with two different TPMS sheet thicknesses by varying laser processing conditions (e.g., contour and hatch conditions in this study). Quasi-static tensile testing of solid samples with corresponding AM conditions and 3-point bending tests of TPMS lattices were performed in accordance with ASTM E8 and ASTM E290, respectively. It was observed that the flexural properties of the 0.75 mm and 0.25 mm TPMS lattices are similar and exhibit different properties with different scan strategies and speed variations under contour-only and hatch-only laser scanning strategies. Also, the 0.75 mm TPMS sheet lattices exhibited 79 % higher flexural stiffness than the 0.25 mm sheet lattices. It was also observed that this observed trend was reversed in the case of tensile properties. Findings from this study can provide new directions towards achieving gradient TPMS lattice designs with varying local mechanical performance by grading the laser scanning strategies to achieve desired mechanical properties and surface topologies.

Triply Periodic Minimal Surface (TPMS) gyroid lattices are mathematically defined surfaces with zero-mean curvature, making them ideal for structures that require high mechanical performance with minimal material use. In this study, TPMS gyroid lattices were analyzed to improve energy absorption by combining density gradients, computational design, simulations, and surrogate modeling using the Gaussian process technique. A total of 29 different lattice configurations were examined, including uniform, linear increase, and linear decrease density gradients, with a range of relative densities ranging from 15% to 80%. The lattices geometries were designed using nTop and tested under quasi-static compressive loading simulations in ANSYS with the properties of Stainless Steel 316L, a common additive manufacturing material. The strain energy at 35% was then extracted to calculate the specific energy absorption (SEA) in J/g. The surrogate model demonstrated a high prediction accuracy of 95.45%, allowing for optimization of density distribution to enhanced energy absorption. The results highlight the benefits of gradient-based TPMS designs over uniform lattices, showing improved performance absorbing total energy. This research supports the development of advanced lattice structures for metal paste deposition additive manufacturing, offering new possibilities for lightweight, smart, engineered and impact-resistant materials. The combination of density gradients and data-driven optimization can be applied to various engineering applications, including aerospace, automotive, and protective gear industries.

In this paper we consider the existence and the compactness of Riemannian metrics of prescribed mean curvature and zero boundary mean curvature on a three dimensional manifold with umbilic boundary (M, g 0 ). We prove that for three dimensional manifolds with umbilic boundaries, which are not conformally equivalent to the three dimensional standard half sphere, any positive function can be realized as the scalar curvature of a Riemannian metric g conformal to g 0 with respect to which the boundary has zero mean curvature. Moreover, all such metrics stay bounded with respect to the C 2,α -topology and in the nondegerate case Morse inequalities hold.

The hyperbolic complex space RH defined by the Clifford algebra and the hyperbolic phase transformation group U4(H) acting on RH are endowed with definite physical meaning in this paper. The hyperbolic complex space RH is isomorphic to the 4-dimensional(4D) Minkowski spacetime, and the hyperbolic phase transformation group U4(H) in RH is just Lorentz transformation group on 4D relativistic spacetime. Furthermore, the general expressions of Lorentz transformation and the velocity transformation on 4D Minkowski spacetime are naturally derived using the composite transformations of the group U4(H). Hence, the well-known special Lorentz transformation in the special relativity(SR) is contained as a special case in our discussions.

Calabi’s Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space L 3 \boldsymbol {L}^3 which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space E 3 \boldsymbol {E}^3 and maximal surfaces in Lorentz-Minkowski space L 3 \boldsymbol {L}^3 , we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in L 3 \boldsymbol {L}^3 which does not admit time-like points ( ( namely, a graph consists of only space-like and light-like points ) ) is a plane.

Treatments of the Lorentz transformation of special relativity at an undergraduate level usually assume that the motion of one observer is along the x axis of another observer, resulting in the (1+1)-dimensional Lorentz transformation group. The (1+n)-dimensional Lorentz group, for n=2 or n=3, is unknown to many physics students because of the simplified (1+1)-dimensional treatments found in most texts. The aim of this article is to simplify the presentation of the (homogeneous, proper, orthochronous) Lorentz group by abstraction to the point where the (1+n)-dimensional Lorentz group can readily be presented to physics students in n space dimensions where n is finite or infinite. The study of the (homogeneous, proper, orthochronous) Lorentz transformation group is simplified and generalized in this article by abstraction, thus obtaining an elegant formalism to deal with the Lorentz group. This new formalism allows one to solve in the abstract Lorentz group previously poorly understood problems in the standard, (1+3)-dimensional Lorentz group. Two such problems, studied in this article, are (i) the problem of determining the Lorentz transformation composition law in a way analogous to the well-understood determination of the Galilean transformation composition law, and (ii) the problem of determining all the Lorentz transformations linking two given points in a Minkowski space. Crucial points in the present study of the abstract Lorentz group are (i) the abstract relativistic velocity addition law; (ii) the abstract Thomas precession, called Thomas gyration; and (iii) the parametrization of the abstract Lorentz transformation by abstract velocity and abstract orientation parameters in such a way that the composition law of abstract Lorentz transformations is given by a corresponding parameter composition law. In the limit of large speed of light c, c→∞, Thomas gyration vanishes, and the Lorentz transformation composition law reduces to the Galilean transformation composition law.

The geometric (or Clifford) algebra Cl3 of three-dimensional Euclidean space is endowed with a natural complex structure on a four-dimensional space. If paravectors, which are formed sums of scalars and vectors, are taken as the “real” elements of the space, then the space can be shown to have a Minkowski spacetime metric, and the paravectors may be identified with spacetime vectors. Physical Lorentz transformations of spacetime vectors are described by spin transformations of the paravectors. The transformation elements are unimodular elements of the algebra, and they form the six-parameter group SL(2,C), the two-fold covering group of restricted Lorentz transformations, SO+(l,3). Its elements are also reducible spinors, whose elements carry a reducible spin representation of SL(2,C). The spin representation is reduced by splitting elements into complementary minimal left ideals. The complex spin space that results has a symplectic structure and its elements belong to Sp(2).KeywordsRest FrameLorentz TransformationGeometric AlgebraSpacetime DiagramObject FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

It is shown that the extended teleparallel gravitational theories, known as $f(T)$ theories, inherit some on shell local Lorentz invariance associated with the tetrad field defining the spacetime structure. We discuss some enlightening examples, such as Minkowski spacetime and cosmological (Friedmann-Robertson-Walker and Bianchi type I) manifolds. In the first case, we show that the absence of gravity reveals itself as an incapability in the selection of a preferred parallelization at a local level, due to the fact that the infinitesimal local Lorentz subgroup acts as a symmetry group of the frame characterizing Minkowski spacetime. Finite transformations are also discussed in these examples and, contrary to the common lore on the subject, we conclude that the set of tetrads responsible for the parallelization of these manifolds is quite vast and that the remnant group of local Lorentz transformations includes one- and two-dimensional Abelian subgroups of the Lorentz group.

The Lorentz group is defined. Special relativity is viewed as the statement that the laws of Physics are invariant under rotations in a four-dimensional space-time. These generalized rotations leave invariant a quadratic form with an indefinite metric, which results in the Lorentz group being non-compact. Its six generators are the ordinary angular momentum J and the boosts N , which are Hermitian in a unitary representation. By identifying the group of proper orthochronous Lorentz transformations with SO0 (3,1) the commutation relations of J and N and the expressions for the two Lorentz Casimirs follow. It is shown the covering group of SO0 (3,1) is SL(2,C). Matrix elements of N are calculated with the help of the Wigner–Eckart theorem and the principal series and complementary series of infinite-dimensional unitary representations is described. Finite-dimensional non-unitary representations are obtained and used to describe the relativistic wave equations of Klein–Gordon, Dirac, Weyl, Proca and Maxwell. Biographical notes on Minkowski, Klein, Gordon, Dirac and Proca are given.

It is well-known that space-like maximal surfaces and time-like minimal surfaces in Lorentz--Minkowski $3$-space $\boldsymbol{R}^{3}_{1}$ have singularities in general. They are both characterized as zero mean curvature surfaces. We are interested in the case where the singular set consists of a light-like line, since this case has not been analyzed before. As a continuation of a previous work by the authors, we give the first example of a family of such surfaces which change type across a light-like line. As a corollary, we also obtain a family of zero mean curvature hypersurfaces in $\boldsymbol{R}^{n+1}_{1}$ that change type across an ($n-1$)-dimensional light-like plane.

Regular tessellations of polygons are not only possible for flat planes (e.g., the {4,4}, {6,3}, and {3,6} tessellations) and the sphere (e.g., the {3,3}, {4,3}, {3,4}, {5,3}, and {3,5} tessellations corresponding to the regular polyhedra), but also for surfaces of negative Gaussian curvature (i.e., hyperbolic planes), of which the {7,3}, {8,3}, and {6,4} tessellations are of greatest actual or potential chemical interest. However, it is not possible to construct an infinite surface with a constant negative Gaussian curvature to accommodate such tessellations because the pseudosphere, the negative curvature “analogue” of the sphere, has an inconvenient cuspidal singularity that prevents it from being used to describe periodic chemical structures. However, patches of varying negative curvature and constant zero mean curvature can be smoothly joined to give various infinite periodic minimal surfaces (IPMSs), which have zero mean curvature and are periodic in all three directions. The unit cells of the simpl...

This paper examines spinor structures and two-component spinor fields in in a class of spacetimes that are space-orientable but not time-orientable. The space-oriented frames form a principal bundle acted on by the group of proper nonorthochronous Lorentz transformations. This group has two double coverings, Sin and Sin, but only Sin acts on the usual two-component spinors associated with Weyl neutrinos in Minkowski space. Consideration is initially restricted to Lorentzian universes-from-nothing, geometries, like antipodally identified deSitter space, that have a single spacelike boundary and a smooth metric with Lorentzian signature. Every such spacetime has a Sin structure, but only a subclass has a Sin structure. Inequivalent Sin- and Sin-spinor structures correspond to members of two classes of homomorphisms from to , where is the orientable double covering of the spacetime manifold M. For general time-nonorientable spacetimes, a similar classification is obtained of Sin structures in terms of homomorphisms from to where E is the bundle of space-oriented frames of M.