Euclidean manifold

Surfaces may found by finite element analysis methods where the curvature of each element of surface is brought iteratively to the correct value. More general energy functions can be imposed in this way. Exact minimal surfaces are merely particular idealizations and their value lies in their being two-dimensional manifolds which have metrics different from that of the euclidean manifold of the plane. 

A more general formulation in terms of a non-euclidean manifold (curved space) has several advantages, the most important of which is a geometrical... 

The role of a boundary in a manifold with boundary can be interpreted with reference to a hyperplane within a Euclidean space E using the concept of halfspace, where the hyperplane is in fact the boundary of the half-space. By appropriate reordering of the coordinates, a half-space Hn becomes the subset of a Euclidean space En containing all points of En with non-negative value for the last coordinate. 

A space M where each point x e M has an open neighborhood homeomorphic to a set open within a Euclidean half-space Hn, is an K-dimensional manifold with boundary. 

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. 

When N > 4 there appears to be too many Zn, since N(N — l)/2 > 3N — 6. However, the Zn are not globally redundant. All Zn are needed for a global description of molecular shape, and no subset of ZN — 6 Zn will be adequate everywhere.49 The space of molecular coordinates which defines the shape of a molecule is not a rectilinear or Euclidean space, it is a curved manifold. It is well known in the mathematical literature that you cannot find a single global set of coordinates for such curved spaces. 

An affine manifold is said to be flat or Euclidean at a point p, if a coordinate system in which the functions Tl-k all vanish, can be found around p. For a cartesian system the geodesics become... 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

Pesic, P. D. (1993) Euclidean hyperspace and its physical significance. Nuovo Cimento B. 108B, ser. 2(10) 1145—53. (Contemporary approaches to quantum field theory and gravitation often use a 4-D space-time manifold of Euclidean signature called hyperspace as a continuation of the Lorentzian metric. To investigate what physical sense this might have, the authors review the history of Euclidean techniques in classical mechanics and quantum theory.)... 

As discussed in Section 9.3, a higher level of mathematical structure is achieved by defining an additional multiplication (X-Y) operation, that is, a rule that associates a (real) scalar with every pair of objects X, Y in the manifold. For Euclidean-like spaces, the scalar product has distributive, commutative, and positivity properties given by... 

It is a remarkable fact that properties (13.4a-c) are necessary and sufficient to give Euclidean geometry. In other words, if any rule can be found that associates a number (say, (X Y)) with each pair of abstract objects ( vectors X), Y)) of the manifold in a way that satisfies (13.4a-c), then the manifold is isomorphic to a corresponding Euclidean vector space. We introduced a rather unconventional rule for the scalar products (X Y) [recognizing that (13.4a-c) are guaranteed by the laws of thermodynamics] to construct the abstract Euclidean metric space Ms for an equilibrium state of a system S, characterized by a metric matrix M. This geometry allows the thermodynamic state description to be considerably simplified, as demonstrated in Chapters 9-12. 

For the reader that would not know about covariant derivative, suppose that one is in the Euclidean case (K = 0) and that the spatial part of the unperturbed manifold has Cartesian coordinates, so that 7ij is simply a Kronecker symbol, and the Vi are simply ordinary derivative with respect to the coordinate xl. 

The universally observed flow of time is another example of a broken symmetry. A theoretical formulation of this proposition is not known, but in principle it should parallel the theory of superconductivity. A high-symmetry state could be associated with Euclidean Minkowski space that spontaneously transforms into a curved manifold of lower symmetry. In this case the hidden symmetry emerges from a Lagrangian which is invariant under the temporal evolution group... 

An elegant but simple model of a five-dimensional universe has been proposed by Thierrin [224]. It is of particular interest as a convincing demonstration of how a curved four-dimensional manifold can be embedded in a Euclidean five-dimensional space-time in which the perceived anomalies such as coordinate contraction simply disappear. The novel proposal is that the constant speed of light that defines special relativity has a counterpart for all types of particle/wave entities, such that the constant speed for each type, in an appropriate inertial system, are given by the relationship... 

Conformal symmetry is very common in nature e.g. we can find it in the nautilus shell and the sunflower. These structures are clearly ordered, even if they do not give sharp diffraction patterns. Here the repetition is non-Euclidean, on a logarithmic spiral (nautilus), or on a torus (sunflower). We are inclined to say that any kind of repetition, conformal or isometric, even in non-Euclidean space, is ordered. However, classification of these more chaotic structures, as for liquids, is less certain. It may be that a liquid can be described as a structure with some of the characteristics of conformal symmetry or perhaps by a representation even more exotic, like a manifold of constant negative curvature. 

The geometrical meaning is that the Eckart subspace pf is perpendicular to the three-dimensional manifold of rigid-body rotation at the reference configuration z.si. The Eckart subspace is Euclidean since the conditions in Eq. (32) are linear. Therefore this space can be spanned by vectors , (p = 1,..., 3m — 6) that specify the 3m 6 directions of (vibrational) normal modes at the reference configuration zVI- in the (3m — 3(-dimensional configuration space. The vectors (m,m are orthonormal as Mi > = [Pg.107]

Let us articulate some technical terminologies here. A rf-dimensional manifold is a set that locally resembles Euclidean space and each point on a manifold must have a neighborhood around itself that looks like a small piece of Rrf. The letters D and O are l-(dimensional) manifolds, but the letters A and X are not. The surfaces of balls and doughnuts are 2-(dimensional) manifolds. In the strict definition of... 

Then what will be the minimum number of dimensions m.( < k) required to embed a d-dimensional manifold A lying in Whitney s embedding theorem [75] states a condition that ensures producing the embedding of the /-dimensional manifold A in Uk onto a reduced state space Rm. Here, in order to capture its essence, let us consider an example of a one-dimensional manifold A, a twisted circle, that will be observed in Um (in 1,2, 3). As shown in Fig. 30, when the 1-manifold A is projected on to a one-dimensional Euclidean space R1, selfintersections (i.e., not one-to-one) occur at almost every point inevitably. For the... 

As was justified in section 3.2.4, the angle a denotes the phase and. 2 the position of the spiral tip. The Palais section coordinate n U is absent here, because the critical spectrum is now three-dimensional, only, and is accounted for by the three-dimensional group SE 2) itself. Therefore the center manifold M. is a graph over the group coordinates e ° ,z) G SE 2). A rigorous derivation of the reduced equation (3.22) has indeed been achieved in [25, 33], under the assumption that the unperturbed spiral wave n (-) is spectrally stabie with the exception of a triple critical eigenvalue due to symmetry see theorem 1. Note that the nonlinearities -y a,z,s) and h a,z,s) obey the lattice symmetry relic of full Euclidean symmetry, namely... 

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