Get An Introduction to Noncommutative Differential Geometry and PDF

By J. Madore

ISBN-10: 0521659914

ISBN-13: 9780521659918

This is often an advent to noncommutative geometry, with particular emphasis on these instances the place the constitution algebra, which defines the geometry, is an algebra of matrices over the advanced numbers. functions to user-friendly particle physics also are mentioned. This moment version is punctiliously revised and contains new fabric on fact stipulations and linear connections plus examples from Jordanian deformations and quantum Euclidean areas. just some familiarity with usual differential geometry and the speculation of fiber bundles is thought, making this e-book available to graduate scholars and newbies to this box.

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Seien (M angende ˜ , g˜) sei vollst¨ Riemannsche Mannigfaltigkeiten gleicher Dimension. (M andig und ˜ , g˜) → (M, g) π : (M ˜ ist dp˜π : Tp˜M ˜ → Tp M eine Isometrie. h. f¨ ur alle p˜ ∈ M ¨ Dann ist π eine Uberlagerung. Beweis. Als Immersion zwischen gleichdimensionalen Mannigfaltigkeiten ist π ein lokaler Diffeomorphismus, also eine lokale Isometrie. Insbesondere bildet π Geod¨atische in Geod¨atische ab. ˜ , p := π(˜ ˜ mit p) und v ∈ Tp M . Dann gibt es v˜ ∈ Tp˜M A) (M, g) ist vollst¨ andig. Seien p˜ ∈ M dπ(˜ v ) = v, und t → π(exp(t˜ v )) ist eine Geod¨atische in (M, g) mit c(0) ˙ = dπ(˜ v ) = v.

Pn x, x >)dσS n−1 1 (Spur A) vol(S n−1 ). n Mit (59) erhalten wir Satz 94 (Satz von Bonnet-Myers: 2. Version). Sei (M, g) eine vollst¨ andige, zusammenh¨ angende Riemannsche Mannigfaltigkeit der Dimension ≥ 2. Es gebe δ > 0, so daß ric(X, X) ≥ δ X f¨ ur alle X ∈ T M . Dann ist M kompakt und π diam(M ) ≤ √ . δ 56 2 15 Der Satz von Synge Satz 95 (Lemma von Synge). Sei (M, g) eine orientierbare Riemannsche Mannigfaltigkeit gerader Dimension mit positiver Schnittkr¨ ummung K > 0. Sei c : [0, L] → M eine glatt-geschlossene Geod¨ atische der L¨ ange L(c) = L > 0.

Beispiel 20. Daraus folgt 1 2 [X, Y ] f¨ ur 1 R(X, Y )Z = − [[X, Y ], Z] 4 und f¨ ur orthonormale X, Y K(σXY ) = 1 [X, Y ] 2 . 4 Man zeigt leicht, daß K f¨ ur SO(3) positiv und konstant ist. F¨ ur SO(n), n > 3 hingegen nimmt K ≥ 0 auch den Wert 0 an. Beispiel 75. F¨ ur dim M =2 ist R(X, Y )Z = K (g(Y, Z)X − g(X, Z)Y ), 42 wobei K : M → R, p → K(Tp M ) die Schnittkr¨ ummung ist. Beweis. Es gen¨ ugt, R(X, Y ) f¨ ur eine ON-Basis (X, Y ) zu kennen. Dazu gen¨ ugt es, R(X, Y )X und R(X, Y )Y zu kennen.

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An Introduction to Noncommutative Differential Geometry and its Physical Applications by J. Madore


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