Re: local classification of riemannian manifolds


[A complimentary Cc of this posting was sent to
harsha
<harsha.v.r@xxxxxxxxx>], who wrote in article <dj4366$4mc$1@xxxxxxxxxxxxxxxxxxxxxxxxx>:
> thanks for the replies, i am currently am reading up on the question.
> couldn't find spivak but berger(a panoramic view of differential
> geometry) has a whole section on this. apparently, curvature does NOT
> determine the metric, indeed he mentions that there are examples of
> diffeomorphisms between two non isometric manifolds M,N which preserve
> the curvature tensor(pull back the curvature on N to the curvature on
> M). i'll look more closely and post later

Just in case it is not explicitly mention in the books you have: if
you fix a point and an orthogonal basis in the tangent space at this
point, then the exponential coordinate system is uniquely defined.
The metric tensor in this coordinate system is an invariant of the
(manifold, point, basis).

So this reduces the question to one how things change when you change
the point and the basis.

Which metric tensors appear in this way? Well, the value at the
origin is given by identity matrix. The only other restriction is
that straight line through the origin are geodesics (with natural
parameter). This gives restrictions on the curvature tensor, thus
some differential equations on the metric tensor.

Hope this helps,
Ilya
.

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