(Pseudo-)Riemannian Metrics
Ironically, in order to tie what we’ve been doing back to more familiar material, we actually have to introduce more structure. It’s sort of astonishing in retrospect how much structure comes along with the most basic, intuitive cases, or how much we can do before even using that structure.
In particular, we need to introduce something called a “Riemannian metric”, which will move us into the realm of differential geometry instead of just topology. Everything up until this point has been concerned with manifolds as “shapes”, but we haven’t really had any sense of “size” or “angle” or anything else we could measure. Having these notions — and asking that they be preserved — is the difference between geometry and topology.
Anyway, a Riemannian metric on a manifold 
is nothing more than a certain kind of tensor field 
of type 
on . At each point 
, the field gives us a tensor:
We can interpret this as a bilinear function which takes in two vectors 
and spits out a number 
. That is, 
is a bilinear form on the space 
of tangent vectors at 
.
So, what makes into a Riemannian metric? We now add the assumption that is not just a bilinear form, but that it’s an inner product. That is, is symmetric, nondegenerate, and positive-definite. We can let the last condition slip a bit, in which case we call a “pseudo-Riemannian metric”. When equipped with a metric, we call a “(pseudo-)Riemannian manifold”.
It’s common to also say “Riemannian” in the case of negative-definite metrics, since there’s little difference between the cases of signature 
and 
. Another common special case is that of a “Lorentzian” metric, which is signature 
or 
.
As we might expect, is called a metric because it lets us measure things. Specifically, since is an inner product it gives us notions of the length and angle for tangent vectors at . We must be careful here; we do not yet have a way of measuring distances between points on the manifold itself. The metric only tells us about the lengths of tangent vectors; it is not a metric in the sense of metric spaces. However, if two curves cross at a point we can use their tangent vectors to define the angle between the curves, so that’s something.
September 20, 2011 -
Posted by John Armstrong |
Differential Geometry, Differential Topology, Geometry, Topology
The Unapologetic Mathematician 
as we move from patch to patch for the manifold, does g (tensor field) need any restrictions? does the form change value depending on which piece of the atlas we’re using?
thanks for this blog series!
That’s the neatest thing: the metric
is defined as a geometric object — a tensor field — so it doesn’t depend on the local coordinate patches at all! All that the patches matter is when you want to represent the inner products with matrices with respect to some basis of the (co)tangent vector space.
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how can it help to our daily life ?
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И здесь
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I like to add that “angle” is interpreted in statistics as “correlation”. So that opens big, big areas of applications and makes “inner product” therefore that much more appealing.