Mast Kalandar

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Sat, 15 Jul 2000

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Question of Riemann

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Here is a question that seems to arise out of reading Riemann's paper on the foundations of geometry. I have been asking this question since I wrote my "Resonance" article on the topic but no one has provided an answer as yet.

Riemann's count for the number of parameters that determine a Riemannian metric on an n dimensional manifold goes as follows.

The metric is given by a symmetric matrix of functions; this correspondes to n(n+1)/2 functions in local co-ordinates. However, this depends on a choice of co-ordinates which is given by n functions. Thus there should be n(n-1)/2 functions on a manifold that determine its geometry.

Riemann then seems to assert that the following provides a possible choice of n(n-1)/2 functions.

Let R be the curvature form of a Riemannian manifold M. We think of R of as a symmetric endomorphism of the second exterior power of the tangent bundle of M. Consider the coefficients of the characteristic polynomial of this endomorphism. These are n(n-1)/2 functions on M.

A natural question that arises is:

Do these determine the Riemmanian structure on M?

In other words if S is the curvature form of another metric on M such that the p_i associated with S are the same as those for R then are the metrics the same?

What Riemann does prove is that if these functions are all zero (and consequently R is zero as well) then the metric is the usual Euclidean metric locally. However, the above question is not answered.

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