From: Boudewijn Moonen
Subject: Intrinsic geometric description of parallel transport
Date: Fri, 26 Feb 1999 18:06:42 +0100
Newsgroups: sci.math.research
For quite a while I have been intrigued by the following question in
Riemannian Geometry. One of the fundamental results is that in any
Riemannian manifold there is an intrinsic notion of parallel transport
along paths. Usually, this notion is obtained by first introducing
algebraically the Levi-Civita-connection \nabla via the Koszul formula
(the ``good old switcheroo'' as Spivak calls it):
\forall X,Y,Z: <\nabla_X Y,Z> :=
1/2 { X + Y + Z - <[X,Y],Z> - <[Y,Z],X> + <[Z,X],Y> }
and then integrating it along paths. The drawback of this approach is
that, though being a fundamental result in geometry, it is substantially
lacking a geometric description. Since parallel transport is an
intrinsic geometric notion, my first question is:
Question 1: Is there an INTRINSIC geometric description of
parallel transport in a general Riemannian manifold?
Results which I am aware of in this direction are the following. Since
a general path can be quite arbitrary, let us content ourselves with a
geometric description of parallel transport along geodesics and then
approximate a general path arbitrarily close by geodesic polygonal
paths.
A) In dimension 2, there IS an intrinsic geometric description of
parallel transport. For this, it suffices to describe how to parallel
transport a sufficiently small tangent vector X along a geodesic
\gamma, from p to q, say. Represent X by a normal geodesic arc \xi
issuing from p of lenght l. Then we have the oriented angle \phi :=
\angle(\gamma,\xi) at p. Then draw at q the geodesic arc \xi' with
\angle(\gamma,\xi') = \phi at q and length l. This geodesic arc
then represents the parallel transport of the vector X along \gamma.
B) It is therfore tempting to proceed in dimensions > 2 as follows. Let
X, p, q and \xi as in A). Since we know how to parallel transport the
tangent vector to a geodesic along the geodesic, we may assume that
X and \dot{\gamma}(0) are linearly independent. Then X and
\dot{\gamma}(0) span a plane in the tangent space and so \xi and
\gamma span, via the exponential mapping, a local surface around p,
which I call the local exponential surface spanned by \xi and \gamma at
p, denoted \exp<\xi,\gamma>_p. Equip this surface with the induced
metric. Then I can parallel transport X within the exponential surface
according to the prescription A), at least if q is sufficiently close
to p. In the general case, subdivide the geodesic from p to q into
sufficiently small arcs and update the local exponential surface at
each subdivision point.
I do not expect that this construction gives in general the
description of parallel transport in the big manifold, since this
requires the constraint that parallel transport along \gamma within the
local exponential surface coincides with the parallel transport along
\gamma in the big manifold.
Question 2: Which are the Riemannian manifolds where this
description of parallel transport hold?
I do expect, however, that this procedure provides a discrete
approximation to the parallel transport in the big manifold in the sense
that it converges to it when making the subdivision arbitrarily small.
I suspect that there are some hints to this in an old paper of Severi's:
F. Severi, Sulla curvatura delle superficie e variet\`a, Rendiconti
del Circolo matematico di Palermo, vol 42 (1917), 227 - 259
but unfortunately my Italian is far too fragmentary as to be able to
unravel the secrets of this paper. Hence
Question 3: Does this procedure converge to the parallel transport
in the big manifold, and is this in Severi or some more recent
source?
C) There is another intrinsic approximation to the description of
parallel transport known as the ``Schild's ladder construction''.
For a description, see
C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation, W. H.
Freeman 1973, Box 10.2, pp 248 - 249
but the discussion there lacks mathematical rigour. Whence
Question 4: Is there a mathematically satisfying justification
somewhere in the literature for the validity of the Schild's
ladder construction?
D) There are EXTRINSIC geometric descriptions of parallel transport
like ``rolling without slipping'' or Cartan's ``development'', which
use the embedding of the manifold in Euclidean space, but I am
interested in intrinsic descriptions. So, finally, a reprise:
Question 5: Are there further known attempts to an intrinsic
geometric description of parallel transport?
--
Dr. Boudewijn Moonen
Institut fuer Photogrammetrie der Universitaet Bonn
Nussallee 15
D-53115 Bonn
GERMANY
e-mail: Boudewijn.Moonen@ipb.uni-bonn.de
Tel.: GERMANY +49-228-732910
Fax.: GERMANY +49-228-732712
==============================================================================
From: bfcedm02@club-internet.fr (Brieuc)
Subject: Re: Intrinsic geometric description of parallel transport
Date: Sat, 27 Feb 1999 00:13:55 GMT
Newsgroups: sci.math.research
On Fri, 26 Feb 1999 18:06:42 +0100, Boudewijn Moonen
wrote:
....
>along paths. Usually, this notion is obtained by first introducing
>algebraically the Levi-Civita-connection \nabla via the Koszul formula
...
>lacking a geometric description. Since parallel transport is an
>intrinsic geometric notion, my first question is:
>
> Question 1: Is there an INTRINSIC geometric description of
> parallel transport in a general Riemannian manifold?
>
a)- abstract connection on a manifold M:
be "u" a vector tangent at "x" to M;
"ux" belongs to the fiber-bundle TM
you want to move x by a little-vector v
x --> "x+v"
how to move u (and ux) ? (transport).
A linear-connection C(x) is the answer:
u --> "u+(C.v).u"
C is a linear appli on u (but also on v).
Now, there is a curve t --> x(t) on M
how to transport u0 tangent at x0 to x1 ?
by means of the diff equa:
find t --> u(t) so that
u' = ( C(xt).x' ).u
starting at u0, x0
where x' = dx/dt & u' = du/dt
b) rieman metric:
at each x, there is a metric g(x)
so that the scalar product is defined by
(u | v) =
g sends linearly a vector on a co-vector,
g is its own dual, and must be an isom.
there should be an unique connection C,
torsion-free, called levi-civita, such that
if you transport u and v with C, you do
not change (u | v) !
some calculus will give (C.u).u :
= u.(g'.v).u/2 - u.(g'.u).v
c) connexion is intrinsic, and geometric:
a geodesic for C is a path t --> x(t)
so that its speed v = x'(t) transports
itself:
v' = (C.v).v
that should coincide with a condition
for the g-shortest path from x0 to x1:
minimal sum of |v|.dt along the path.
this cond. can be reformulated with:
the nabla (covariant derivative along v) :
for any path t --> u(t) x(t) with v = x'(t)
nabla = u' - (C.v).u
for a geodesic, u = v, and nabla = 0
d) it is convenient to help with the
rectangular commutative diagram
cvux -------> vx
| |
| |
ux -------> x
where ux and vx are in TM, at x,
and cvux in T(TM), at ux.
vx -->x & ux-->x are foot-projections,
cvux-->ux , too, but cvux-->vx is
the derivative of uv-->x !
"geometrie differentielle intrinseque"
P. Malliavin, Hermann. Paris
"elements d'analyse" tomes 3 & 4
J.Dieudonne, Gauthier-Villars Paris.
hope these heuristics helps you.
Brieuc.