1
$\begingroup$

A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a constant... But I would like to know the more general meaning/definition of the term. Thanks.

Added: In particular, I would like to know what form such a R metric would look like e.g. $ds^2=adx^2+bdy^2$

P.S. I have tried searching for a definition, but I couldn't find a proper definition.

  • 0
    @Neal: Oops, sorry. I meant that it is always possible to choose coordinates so that the metric is _diagonal_.2012-04-30

1 Answers 1

1

Definitions can be found at: http://en.wikipedia.org/wiki/Locally_Euclidean.

I understand that Wikipedia may not be your favorite source, so if you can access googlebooks, searching for "locally euclidean" yields at least 6 out of the top 10 hits defined it as "each point has a neighborhood homeomorphic to $\mathbb{R}^n$." As previous commentors have also stated, sometimes the requirement for homeomorphisms is strengthened to require diffeomorphisms instead.