By Yuri Gliklikh
This booklet offers a typical remedy to 3 parts of software of worldwide research to Mathematical Physics formerly thought of particularly far away from one another. those components are the geometry of manifolds utilized to classical mechanics, stochastic differential geometry utilized in quantum and statistical mechanics, and infinite-dimensional differential geometry primary for hydrodynamics.
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Extra resources for Global Analysis in Mathematical Physics: Geometric and Stochastic Models (Applied Mathematical Sciences)
Sample text
On the other hand, coordinate representations are often a good theoretical tool for abstract calculations. 1. 2. r; Â/. 3. Any left-invariant metric on a Lie group G can be written as gD. 1 2 / C C. n 2 / 16 1 Riemannian Metrics using a coframe dual to left-invariant vector fields X1 ; : : : ; Xn forming an orthonormal basis for Te G. If instead we just begin with a frame of left-invariant vector fields X1 ; : : : ; Xn and dual coframe 1 ; : : : ; n , then a left-invariant metric g depends only on its values on Te G and can be written as g D gij i j , where gij is a positive definite symmetric matrix with real-valued entries.
0; 4/ version. In many texts it is placed first. Our choice appears natural given how we write these tensors in invariant notation in chapter 3. 5 Some Tensor Concepts 29 R D Rkl ij Ek ˝ El ˝ ˝ i D Rlijs gsk Ek ˝ El ˝ i j ˝ j : Here we must be careful as there are several different possibilities for raising and lowering indices. 2; 2/-tensors. The way we did it gives what we will call the curvature operator. 2 Contractions Contractions are traces of tensors. T/ D trT D Tii : An instructive example comes from considering the rank 1 tensor X ˝ !
X1 ; : : : ; xn / on an open set U of M we can thus construct bilinear forms dxi dxj . @i ; @j / are denoted by gij . This gives us a representation of g in local coordinates as a positive definite symmetric matrix with entries parametrized over U. Initially one might think that this gives us a way of concretely describing Riemannian metrics. That, however, is a bit optimistic. Just think about how many manifolds you know with a good covering of coordinate charts together with corresponding transition functions.