Monomialization of singular metrics on real surfaces

Abstract

Let B be a real analytic vector bundle of rank 2 over a smooth real analytic surface S, equipped with a real analytic fiber-metric g and such that there exists a real analytic mapping of vector bundles T S → B inducing an isomorphism outside a proper sub-variety of S. Let κ be a real analytic 2-symmetric tensor field on B. Our main result, Theorem 9.2, roughly states the following: There exists a locally finite composition of points blowings-up σ : S → S such that there exists a unique pair of real analytic singular foliations F1 and F2 on Sonly with simple singularities adapted to the exceptional divisor E and orthogonal for the (regular extension of the) pull back on S of the fiber-metric g (only semi-positive definite along E) -locally simultaneously diagonalizing the pull-back on S of the original 2-symmetric tensor field κ.When S is the resolved surface of an embedded resolution of singularities of an embedded real analytic surface singularity S0 our result thus yields a local presentation of the extension h of the pull-back on S of the inner-metric of S0 at any point of the exceptional divisor E. We furthermore recover that the pulled-back inner metric h is locally quasi-isometric to the sum of the (symmetric tensor) square of differentials of (independent) monomials in the exceptional divisor E, namely, Hsiang & Pati property is satisfied at every point of the resolved surface S.CONTENTS 10.4. Local normal form for the induced metric 32 References 34such that there exists, up to permutation, a pair of O X 2 -invertible sub-modules Θ 1 and Θ 2 of Ω 1 X 2 , such that i) Each foliation Θ i admits only simple singularities adapted to E 2 . ii) Every point a 2 of X 2 admits an open neighborhood U 2 ∋ a 2 such that if σ 2 (a 2 ) is not a singular point of X 0 , then (Dσ 2 (a 2 ))(ker Θ 1 (a 2 )) and (Dσ 2 (a 2 ))(ker Θ 2 (a 2 )) are orthogonal lines of T σ 2 (a 2 ) X 0 ;iii) The pull-back of the metric g 0 | X 0 by the resolution mapping σ 2 extends on X 2 as a real analytic semi-Riemannian metric g 2 , which writes on U 2 asOf course what is also important is point iv) of the main theorem, since we unexpectedly (that is without further blowings-up) obtain the following (Corollary 10.9 and Corollary 10.12):Corollary. Under the hypotheses of Theorem 1.2 each point a 2 in E 2 admits Hsiang & Pati coordinates, namely there exists local analytic coordinates (u, v) centered at a 2 such that i) If a 2 is a smooth point of E 2 , there exists local coordinates (u, v) at a 2 such that (E 2 , a 2 ) = {u = 0}, then the (extension of the) pulled-back metric g 2 is quasi-isometric (nearby a 2 ) to the metric given byfor non-negative integer numbers l ≥ k.ii) If a 2 is a corner point of E 2 , there exists local coordinates (u, v) at a 2 such that (E 2 , a 2 ) = {uv = 0}, then the (extension of the) pulled-back metric g 2 is quasi-isometric (nearby a 2 ) to the metric given by

Blow-analytic equivalence versus contact bi-Lipschitz equivalence

“…In points (i) and (ii) of Lemma 4.1, the local coordinates (u, v) at a are called Hsiang & Pati coordinates at a (see [13,10,2,9]). Following Remark 4.2 we introduce the following Hsiang & Pati data allows to describe, up to local quasi-isometry (see also [13,10,2]) the local form at any point point of the exceptional locus E of the pull-back of the Euclidean metric by the "resolution mapping π".…”

Bisimplicial Commutative Algebras and Crossed Squares

“…It can be seen the role of these pairings for the relations among bisimplicial groups and crossed squares. Arvasi and Porter [13], using the Carrasco and Cegarra pairing operators for a Moore complex in a simplicial (commutative) algebra, and they have defined the functions C α,β functions, and as an application, they proved that the category of 2-crossed modules of commutative algebras introduced by Grandjeán and Vale in [1] is equivalent to that of simplicial commutative algebras with Moore complex of length 2. Of course, this is the commutative algebra version of Conduché's result [2].…”