By J. Madore

This can be an creation to noncommutative geometry, with particular emphasis on these situations the place the constitution algebra, which defines the geometry, is an algebra of matrices over the advanced numbers. functions to easy particle physics also are mentioned. This moment version is punctiliously revised and contains new fabric on truth stipulations and linear connections plus examples from Jordanian deformations and quantum Euclidean areas. just some familiarity with traditional differential geometry and the idea of fiber bundles is believed, making this ebook obtainable to graduate scholars and newbies to this box.

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Since we are not assuming that readers of this book have extensive familiarity with topological ideas, we will not be able to give complete proofs of some of the global results about curves that we discuss in this chapter. 1 Simple closed curves In this chapter, we shall consider plane curves of the following type. 1 A simple closed curve in R2 is a closed curve in R2 that has no self-intersections. 1007/978-1-84882-891-9 3, c Springer-Verlag London Limited 2010 ⃝ 55 56 3. , it is contained inside a circle of sufficiently large radius.

2), so the intermediate value theorem tells us that, if κs takes both the value κ and the value −κ, it must take all values between. Thus, either κs = κ at all points of the curve, or κs = −κ at all points of the curve. In particular, κs is constant. The idea now is to show that, whatever the value of κs , we can find a parametrized circle whose signed curvature is κs . The theorem then tells us that every curve whose signed curvature is κs can be obtained by applying a direct isometry to this circle.

2), so the intermediate value theorem tells us that, if κs takes both the value κ and the value −κ, it must take all values between. Thus, either κs = κ at all points of the curve, or κs = −κ at all points of the curve. In particular, κs is constant. The idea now is to show that, whatever the value of κs , we can find a parametrized circle whose signed curvature is κs . The theorem then tells us that every curve whose signed curvature is κs can be obtained by applying a direct isometry to this circle.

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