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- W2006793956 abstract "1. Introduction. A straight line-segment (the basis of the fundamental entity of classical vector analysis) is perhaps the simplest nontrivial example of a continuous flat space. Furthermore, it seems that, from an elementary point of view, a higher space should be regarded as βat only if it is composed of these elementary flat spaces (strokes) in such a way that the vector Δp joining any two points of the higher space lies entirely in that space. A criterion of similar spirit is the requirement that no such Δp have a component normal to the space. Since we prefer to regard a plane patch with a concavity in the boundary curve as being completely flat, we adopt (for heuristic purposes) the second test. Either view leads quite naturally to the investigation of certain immersed spaces RN from the standpoint of the various derivatives of the radius vector p since these local quantities determine Δp. With regard to the space Rjy, we assume that it is a Riemannian space defined vectorially by means of the equation with p the radius-vector from an origin in an enveloping classical vector-space to a generic point of R^. The function p (x) is assumed to be such that Δp can be determined by a Taylor's expansion. Obviously, the term βat as used here is applicable to straight lines and planes but not to certain intrinsically βat spaces (i.e., flat in the sense that the Riemann-Christoffel tensor vanishes), such as curved lines and cylinders. The basic datum here is the Euclidean vector-space with it directed strokes which can be used to test lines and other immersed spaces for complete flatness. Since Δp is determined by the various derivatives of p at the point of issue of Δp, it is obvious that complete βatness or deviation therefrom will show up in the derivatives of p. These vector derivatives may have components normal to RN and hence, in this case, do not belong to RJSJ in a strong sense. However, by forming certain dot products, they can be converted to scalar functions which may more appropriately by assigned to the immersed space. Among these" @default.
- W2006793956 created "2016-06-24" @default.
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- W2006793956 date "1953-03-01" @default.
- W2006793956 modified "2023-09-26" @default.
- W2006793956 title "On certain metric extensors" @default.
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- W2006793956 doi "https://doi.org/10.2140/pjm.1953.3.25" @default.
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