Differential Geometry

Elements of Noncommutative Geometry (Birkhauser Advanced by Joseph C. Varilly, Hector Figueroa, Jose M. Gracia-Bondia PDF

By Joseph C. Varilly, Hector Figueroa, Jose M. Gracia-Bondia

ISBN-10: 0817641246

ISBN-13: 9780817641245

This quantity covers a variety of subject matters together with assets of noncommutative geometry; basics of noncommutative topology; K-theory and Morita equivalance; noncommutative integrodifferential calculus; noncommutative Riemannian spin manifolds; commutative geometrics; tori; facts of Connes' Riemannian spin theorem; moment quantization; noncommutative quantum box thought; C*-algebras; Hopf algebras; Clifford algebras; Moyal algebras; pseudodifferential operators; the commutative Chern personality theorem.

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Extra resources for Elements of Noncommutative Geometry (Birkhauser Advanced Texts Basler Lehrbucher)

Example text

6) ϑ = arccos x3 / ((x1 )2 + (x2 )2 + (x3 )2 ,      ϕ = arctan(x2 /x1 ). Coordinate lines of spherical coordinates form three families. The first family is composed of the rays coming out from the point O; the second family is formed by circles that lie in various vertical planes passing through the axix Ox3 ; and the third family consists of horizontal circles whose centers are on the axis Ox3 . Exactly three coordinate lines pass through each regular point of the space E, one line from each family.

However, for ∇k the special name is reserved, it is called the operator of covariant differentiation or the covariant derivative. Below (in Chapter III) we shall see that the concept of covariant derivative can be extended so that it will not coincide with the partial derivative any more. Let A be a differentiable tensor field of the type (r, s) and let X be some arbitrary vector field. Let’s consider the tensor product ∇A ⊗ X. This is the tensor field of the type (r + 1, s + 1). The covariant differentiation adds one covariant index, while the tensor multiplication add one contravariant index.

Jm−1 n−1 k jn ... js−1 k=1 q=1 ∂xq proves the fifth property. This completes the proof of the theorem in whole. § 6. The metric tensor and the volume pseudotensor. Let O, e1 , e2 , e3 be some Cartesian coordinate system in the space E. The space E is equipped with the scalar product. Therefore, the basis e1 , e2 , e3 of any Cartesian coordinate system has its Gram matrix gij = (ei | ej ). 1) The gram matrix g is positive and non-degenerate: det g > 0. 2) follows from the Silvester criterion (see [1]).

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Elements of Noncommutative Geometry (Birkhauser Advanced Texts Basler Lehrbucher) by Joseph C. Varilly, Hector Figueroa, Jose M. Gracia-Bondia

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