 By K. Kenmotsu

Best differential geometry books

An Introduction to Noncommutative Geometry

Noncommutative geometry, encouraged via quantum physics, describes singular areas by means of their noncommutative coordinate algebras and metric buildings through Dirac-like operators. Such metric geometries are defined mathematically by way of Connes' concept of spectral triples. those lectures, brought at an EMS summer time university on noncommutative geometry and its functions, supply an outline of spectral triples in keeping with examples.

Geometry, Topology and Quantization

This can be a monograph on geometrical and topological positive factors which come up in a variety of quantization techniques. Quantization schemes ponder the feasibility of arriving at a quantum procedure from a classical one and those contain 3 significant methods viz. i) geometric quantization, ii) Klauder quantization, and iii) stochastic quanti­ zation.

Complex Spaces in Finsler, Lagrange and Hamilton Geometries

From a old viewpoint, the idea we undergo the current learn has its origins within the recognized dissertation of P. Finsler from 1918 ([Fi]). In a the classical concept additionally traditional category, Finsler geometry has in addition to a few generalizations, which use an identical paintings strategy and that are thought of self-geometries: Lagrange and Hamilton areas.

Introductory Differential Geometry For Physicists

This booklet develops the math of differential geometry in a much more intelligible to physicists and different scientists drawn to this box. This publication is largely divided into three degrees; point zero, the closest to instinct and geometrical event, is a quick precis of the speculation of curves and surfaces; point 1 repeats, reviews and develops upon the conventional equipment of tensor algebra research and point 2 is an advent to the language of contemporary differential geometry.

Extra resources for Differential Geometry of Submanifolds

Example text

The fundamental objects are generalizations of vectors called tensors. Tensors can be viewed from many different perspectives. Mathematicians introduce tensors formally as a quotient of a certain module, while physicists introduce tensors using objects with many indices that transform in a specific way under a change of basis. Each definition is viewed disparagingly by the other camp, even though they are equivalent and both have their uses. We follow a middle approach here. 1 The tensor product To start, we define a new kind of vector product, called the tensor product, usually denoted by the symbol ⊗.

18) while the antisymmetric part Tasym of T is the antisymmetric tensor with components (Tasym )i j := 1 Ti j − T ji . 19) Evidently, for a (0, 2) tensor, T is the sum of its symmetric and antisymmetric parts. These ideas can be generalized to higher-order tensors, but the results are not as simple. Defining what is meant by “definite symmetry” for higher-order tensors requires some understanding of the representation theory of the symmetric group, which is beyond the scope of this work. Fortunately, for most purposes we need only consider the higher-order analogues of symmetric and antisymmetric tensors, and these are easy enough to describe using basic ideas of permutation theory, as follows.

Then we μ → λ ∧ μ. 57) for some linear functional f λ on n− p V . Given an inner product g, the Riesz lemma guarantees the existence of a unique (n − p)-vector λ such that g( λ, μ) = f λ (μ). 58) n− p The element λ ∈ V is called the Hodge dual or Hodge star of λ. 58) we may write λ ∧ μ = g( λ, μ)σ. 59) Written this way it is clear that the Hodge dual depends on both the inner product g and the choice of basis element σ for n V . At first sight the definition of the Hodge dual appears to be worthless, because it asserts the existence of an object but seems to give you no means of computing it.