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.

Show description

Read or Download An Introduction to Noncommutative Differential Geometry and its Physical Applications PDF

Similar differential geometry books

Joseph C. Varilly's An Introduction to Noncommutative Geometry PDF

Noncommutative geometry, encouraged by means of quantum physics, describes singular areas by way of their noncommutative coordinate algebras and metric constructions through Dirac-like operators. Such metric geometries are defined mathematically by means of Connes' conception of spectral triples. those lectures, brought at an EMS summer time tuition on noncommutative geometry and its purposes, supply an summary of spectral triples in response to examples.

Download PDF by Michèle Audin: Torus Actions on Symplectic Manifolds

This can be a longer moment version of "The Topology of Torus activities on Symplectic Manifolds" released during this sequence in 1991. the fabric and references were up to date. Symplectic manifolds and torus activities are investigated, with a variety of examples of torus activities, for example on a few moduli areas.

Additional resources for An Introduction to Noncommutative Differential Geometry and its Physical Applications

Sample text

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.

Download PDF sample

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

by Jason

Rated 4.08 of 5 – based on 44 votes