Affine Geometries of Paths Possessing an Invariant Integral by Eisenhart L. P.

By Eisenhart L. P.

Show description

Read Online or Download Affine Geometries of Paths Possessing an Invariant Integral PDF

Best geometry and topology books

Low-dimensional geometry: From Euclidean surfaces to hyperbolic knots

The examine of three-dimensional areas brings jointly components from a number of parts of arithmetic. the main striking are topology and geometry, yet parts of quantity conception and research additionally make appearances. some time past 30 years, there were awesome advancements within the arithmetic of three-dimensional manifolds.

Additional resources for Affine Geometries of Paths Possessing an Invariant Integral

Example text

Let A be an affine C-algebra and M ∈ repn A. M is Zariski open in its closure O(M ). M . then, by the above remark Consider the ’orbit-map’ GLn O(M ) = φ(GLn ) contains a Zariski open subset U of O(M ) contained in the image of φ which is O(M ). U is also open in O(M ). Next, we claim that O(M ) contains a closed orbit. Indeed, assume O(M ) is not closed, then the complement CM = O(M ) − O(M ) is a proper Zariski closed subset whence dim C < dim O(M ). But, C is the union of GLn -orbits O(Mi ) with dim O(Mi ) < dim O(M ).

In particular, we apply this to the underlying vectorspace of Mψ which is V = Cn (column vectors) on which GLn acts by left multiplication. We define Mj = ⊕i>j Vλ,i and claim that this defines a finite filtration on Mψ with associated graded A-module Mρ . For any a ∈ A (it suffices to vary a over the generators of A) we can consider the linear maps φij (a) : Vλ,i ⊂ a. ✲ V = Mψ ✲ Mψ = V ✲ ✲ Vλ,j (that is, we express the action of a in a blockmatrix Φa with respect to the decomposition of V ). λ(t)−1 with corresponding blocks Vλ,i φij (a) ✲ Vλ,j ✻ λ(t)−1 Vλ,i that is φij (a) = t j−i λ(t) ❄ ✲ Vλ,j φij (a) φij (a).

D} and let σ = (i1 i1 . . iα )(j1 j2 . . jβ ) . . (z1 z2 . . zζ ) be a decomposition of σ ∈ Sd into cycles including those of length one. The map T assigns to σ a formal necklace Tσ (x1 , . . , xd ) defined by Tσ (x1 , . . , xd ) = t(xi1 xi2 . . xiα )t(xj1 xj2 . . xjβ ) . . t(xz1 xz2 . . (v1 ⊗ . . ⊗ vd ) = vσ(1) ⊗ . . ⊗ vσ(d) hence determines a linear map λσ ∈ End(V ⊗d ). Recall from section 3 that under the natural identifications (Mn⊗d )∗ (V ∗⊗d ⊗ V ⊗d )∗ End(V ⊗d ) the map λσ defines the multilinear map µσ : Mn ⊗ .

Download PDF sample

Rated 4.80 of 5 – based on 46 votes