Read e-book online Basic Elements of Differential Geometry and Topology PDF

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By S.P. Novikov, A.T. Fomenko

ISBN-10: 0792310098

ISBN-13: 9780792310099

One provider arithmetic has rendered the 'Et moi, ..., si j'avait su remark en revenir, je n'y serais element aile.' human race. It has positioned good judgment again Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded n- sense'. The sequence is divergent; for that reason we are able to do anything with it. Eric T. Bell O. Heaviside Matht"natics is a device for idea. A hugely useful software in an international the place either suggestions and non linearities abound. equally, all types of elements of arithmetic seNe as instruments for different components and for different sciences. employing an easy rewriting rule to the quote at the correct above one reveals such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One carrier common sense has rendered com puter technological know-how .. .'; 'One provider type thought has rendered arithmetic .. .'. All arguably precise. And all statements available this fashion shape a part of the raison d'etre of this sequence.

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Extra info for Basic Elements of Differential Geometry and Topology (Mathematics and its Applications)

Example text

Gij =gij(zl, .... z~. A metric is said to be Euclidean if there exist new coordinates ;XI. ~ =;(zl..... z"). Id~Jazl ¢ O. such that: r ..... Relative to coordinates r ..... ;XI we have: 29 RIEMANNIAN METRIC IN EUCLIDEAN SPACE r I, gij = 50 = ~l 0' i =i, l . }, and the coordinates xl, ... ,;tJ are tenned Euclidean coordinates. We always require mat the determinant Igijl be non-zero or, in omer words, that the metric gij be non-degenerate. z'I» If the matrix (giJ{zl, ... , determines a positive quadratic form- that is, the lengths of all non-zero vectors (and, therefore, of all curve segments) are positive, then we say that go represents the Riemannian metric.

We assume that Iwl ¢ 0 and Ivl ¢ 0; such points are called non-degenerate points of the curve. We assume here Ivl I, wv 0 (or w J.. v). Consider the vector b [v, nJ. n wllwl. We shall call b the vector of binormal to the curve or the binormal to the curve, and n the vector of the principal normal to the curve. or the principal normal). We can readily see that: = = = Ibl = Ivlln IIsin $1 = I, = b J.. v, b J.. n. We thus have an ortbononnal frame (v, n, b) at each point of the curve where Iwl ¢ 0 (ie.

The group preserving the pseudo-Euclidean metric is the group of hyperbolic rotations (Figure 8). W which consisted of two connected components (two pieces) - two circumferences. The group of hyperbolic rotations has a more complicated organization: it consists of four connected components (four pieces): {(~~~ :n (:::~ ::~~~); '1'). ch 'I' - sh ( sh 'V - ch 'I' • Each of these pieces is homeomozphic to a real straight line IR 1. '1')11· - ch 'I' sh ( - sh 'I' ch'l' 39 PSEUDO·EUCLIDEAN SPACE AND LOBACHEVSKY GEOMETRY Figure 8.

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Basic Elements of Differential Geometry and Topology (Mathematics and its Applications) by S.P. Novikov, A.T. Fomenko


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