Euclidean tangent space

In om pseudo-Euclidean tangent space it is customary to distinguish between time-hke, space-like and light-like events, well aware that this is another caricatme of curved space-time. For convenience, space-like events are usually ignored as physically unreal. There is no justification for this assumption in four-dimensionally curved space-time. 

This cone is real in the case of relativity theory, while the quadratic form gij is indefinite. Prom the point of view stressed by E. Cartan (bibb 1928,1) the Riemannian geometry of the underlying world is to be considered as the theory of these connected Euclidean tangent spaces. The generalization that we have in mind now consists of the following ... 

Rather than a Euclidean geometry we have, a non-Euclidean geometry in each tangent space, with our quadric as absolute plane in the Cayley sense. Our new geometry therefore is the overall theory of this set of Cayley spaces, in the same way that Riemannian geometry was the theory of the Euclidean tangent spaces of the underlying space. 

It has been argued [8] that the transformation from curved space-time to Euclidean tangent space is described by the golden ratio. This is not an entirely unexpected conclusion, in view of the prominence of r in the operation of selfsimilar symmetries, related to equiangular logarithmic spirals. 

The findings reported here provide new evidence for the unity of micro- and macrophysics and refute the perception of separate quantum and classical domains. The known universe exists as a four-dimensional space-time manifold but is observed in local projection as three-dimensional Euclidean tangent space that evolves in universal time. The observable world, at either micro- or macroscale, can be described in either four-dimensional (nonclassical) or in classical three-dimensional detail. The descriptive model may change, but the reality stays the same. This realization is at the root of self-similarity between large and small. The symmetry operator, which reflects the topology of space-time, is the golden logarithmic spiral. 

To specify the directions of two different vectors at nearby points it is necessary to define tangent vectors at these points. Stated in different terms, at each point of space-time, known as the contact point, there is an associated tangent Minkowski space. The theory of these spaces together with the underlying space becomes a Riemannian geometry if a Euclidean metric is introduced in each tangent space by means of a differential quadratic form. ... 

The generalized non-Euclidean geometry sketched in chapter I is the theory of a set of tangent spaces that each contains a quadric. The theory of a quadric finds its most satisfactory form in the realm of common projective... 

Each Riemann metric is known to produce a Euclidean metric in each tangent space. With respect to the mode of measurement (3), the surface... 

An application of (27) is the proposition that the non-Euclidean distance of two points and of a tangent space is conserved on displacement A. The non-Euclidean distance of both points is determined by... 

The mapping (8) of the five-dimensional space to the tangent space hence implies some sort of perpendicular projection of the Euclidean space with metric 7 /3 on the Euclidean plane with metric 5. ... 

The differential that characterizes non-Euclidean space is known as curvature (Lee, 1997). In two dimensions it describes how a smooth curve deviates from linearity. The curvature, which varies from point to point, is specified in terms of the osculating circle of radius R and centred on the perpendicular to the tangent at p, and which follows the curve in the vicinity of p. On an infinitesimal scale each point on a curve has a unique osculating circle. 

Here, we briefly overview the basic definitions and relations used to describe curvilinear coordinate systems in Euclidean space. These definitions are used to derive the governing equations in Section 5.4. The kinematics of the membrane is also expressed in differential geometry. For further discussion on the topic refer to Carmo [17] and Kreyszig [18]. A two-dimensional surface 5 is characterized by a general set of coordinates as shown in Figure 5.1. The point ( k in the parameter domain V and its mapping x on the surface 5 are defined by the vector x = Jt( k % )-The associated tangent vectors read... 

Casual interpretation of the local environment as three-dimensional space and universal time flow is not consistent with the known four-dimensional structure of space-time on a cosmic scale. Local Euclidean space is said to be tangent to the underlying four-dimensional curved space-time. 

A common approximation that reduces the equation into a three-dimensional wave equation assumes the separation of space and time coordinates, which is the basis of wave mechanics. For many purposes, this is a good approximation in tangent Euclidean space, but it has no validity in curved four-dimensional space-time. 

It may be unexpected to find that number theory and traditional wave mechanics yield comparable reconstructions of extranuclear electronic configurations. However, both models are based on classical waves in three-dimensional space, appropriate for the understanding of atomic structure in tangent Euclidean space. 

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