Torsors

From: Bossavit (bossavit_at_lgep.supelec.fr)
Date: 09/22/04 

Date: Wed, 22 Sep 2004 18:41:06 +0200

In "Torsors made easy",

http://math.ucr.edu/home/baez/torsors.html

John Baez presents the concept of "principal homogeneous space"
(transitive group action with stabilizer reduced to identity), a.k.a.
"torsor". A torsor, in his words, is "like a group that has forgotten
its identity", a notion which is important enough, indeed, to deserve a
shorter name than "principal homogeneous space".

 In the French tradition, we call "torseur", the English equivalent of
which would be "torsor", an "equiprojective vector field" in 3D space,
i.e., a field x --> V(x) such that V(y) - V(x) is orthogonal to y -
x. Such a field can be characterized by two vectors (a polar one and an
axial one), and can be interpreted as, e.g., the velocity field of a
moving solid. "Torseurs", in this acception, can therefore be seen as
elements of (a representation of) a Lie algebra, here the Lie algebra of
the 3D displacement group. Our "torseurs" thus look very much like
tangent vectors to a "torsor" manifold. Not too remote a concept, but
definitely a different one.

 Puzzled by this apparent cultural gap, I have searched the web for
"equiprojective field", "equiprojectivity", etc., and found, to my
surprise, that this concept of equiprojective vector field seems to be
ignored in the Anglo-Saxon world. (When the words themselves can be
found, it's in English papers by French-speaking authors.)

 Yet, the concept must appear in the curriculum as some stage, in
kinematics, dynamics of solid bodies, etc. How do you call it?

[Moderator's remark: in French the word torseur is of course also
used in Baez's sense, for a principal homogeneous space]