Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 21st 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexcross-linked with _[[Euler form]]_ and added these pointers: *** Discussion of _[[Euler forms]]_ ([[differential form]]-representatives of Euler classes in [[de Rham cohomology]]) as [[Pfaffians]] of [[curvature forms]]: * {#MathaiQuillen86} [[Varghese Mathai]], [[Daniel Quillen]], below (7.3) of _Superconnections, Thom classes, and equivariant differential forms_, Topology Volume 25, Issue 1, 1986 (<a href="https://doi.org/10.1016/0040-9383(86)90007-8">10.1016/0040-9383(86)90007-8</a>) * {#Wu05} Siye Wu, Section 2.2 of _Mathai-Quillen Formalism_, pages 390-399 in _Encyclopedia of Mathematical Physics_ 2006 ([arXiv:hep-th/0505003](https://arxiv.org/abs/hep-th/0505003)) * [[Hiro Lee Tanaka]], _Pfaffians and the Euler class_, 2014 ([pdf](http://www.hiroleetanaka.com/pdfs/2014-fall-230a-lecture-26-gauss-bonnet-chern.pdf)) * {#Nicolaescu18} [[Liviu Nicolaescu]], Section 8.3.2 of _Lectures on the Geometry of Manifolds_, 2018 ([pdf](https://www3.nd.edu/~lnicolae/Lectures.pdf), [MO comment](https://mathoverflow.net/a/117982/381)) <a href="https://ncatlab.org/nlab/revision/diff/Pfaffian/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/Pfaffian/11">v11</a>, <a href="https://ncatlab.org/nlab/show/Pfaffian">current</a>

   cross-linked with Euler form and added these pointers:

   ---

   Discussion of Euler forms (differential form-representatives of Euler classes in de Rham cohomology) as Pfaffians of curvature forms:

   • {#MathaiQuillen86} Varghese Mathai, Daniel Quillen, below (7.3) of Superconnections, Thom classes, and equivariant differential forms, Topology Volume 25, Issue 1, 1986 (10.1016/0040-9383(86)90007-8)

   • {#Wu05} Siye Wu, Section 2.2 of Mathai-Quillen Formalism, pages 390-399 in Encyclopedia of Mathematical Physics 2006 (arXiv:hep-th/0505003)

   • Hiro Lee Tanaka, Pfaffians and the Euler class, 2014 (pdf)

   • {#Nicolaescu18} Liviu Nicolaescu, Section 8.3.2 of Lectures on the Geometry of Manifolds, 2018 (pdf, MO comment)

   diff, v11, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the actual statement that $Pf^2 = det$ ([here](https://ncatlab.org/nlab/show/Pfaffian#PfaffianIsSquareRootOfDeterminant)) and a pointer to explicit proofs in * Howard E. Haber, _Notes on antisymmetric matrices and the pfaffian_, 2015 ([pdf](http://scipp.ucsc.edu/~haber/webpage/pfaffian2.pdf)) I gather a proof is also spelled out somewhere in Spivak's _Comprehensive Introduction to Differential Geometry_ but I haven't tracked down the section/page number yet. <a href="https://ncatlab.org/nlab/revision/diff/Pfaffian/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Pfaffian/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Pfaffian">current</a>

   added the actual statement that $Pf^2 = det$ (here) and a pointer to explicit proofs in

   • Howard E. Haber, Notes on antisymmetric matrices and the pfaffian, 2015 (pdf)

   I gather a proof is also spelled out somewhere in Spivak’s Comprehensive Introduction to Differential Geometry but I haven’t tracked down the section/page number yet.

   diff, v15, current