The Unapologetic Mathematician

(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 $M$ is nothing more than a certain kind of tensor field $g$ of type $(0,2)$ on $M$. At each point $p\in M$, the field $g$ gives us a tensor:

$\displaystyle g_p\in\mathcal{T}_p^*M\otimes\mathcal{T}_p^*M\cong\left(\mathcal{T}_pM\otimes\mathcal{T}_pM\right)^*$

We can interpret this as a bilinear function which takes in two vectors $v_p,w_p\in\mathcal{T}_pM$ and spits out a number $g_p(v_p,w_p)$. That is, $g_p$ is a bilinear form on the space $\mathcal{T}_pM$ of tangent vectors at $p$.

So, what makes $g$ into a Riemannian metric? We now add the assumption that $g_p$ is not just a bilinear form, but that it’s an inner product. That is, $g_p$ is symmetric, nondegenerate, and positive-definite. We can let the last condition slip a bit, in which case we call $g$ a “pseudo-Riemannian metric”. When equipped with a metric, we call $M$ 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 $(n,0)$ and $(0,n)$. Another common special case is that of a “Lorentzian” metric, which is signature $(n-1,1)$ or $(1,n-1)$.

As we might expect, $g$ is called a metric because it lets us measure things. Specifically, since $g_p$ is an inner product it gives us notions of the length and angle for tangent vectors at $p$. We must be careful here; we do not yet have a way of measuring distances between points on the manifold $M$ 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 $p$ we can use their tangent vectors to define the angle between the curves, so that’s something.

September 20, 2011 -

1. 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!

Comment by scot | September 21, 2011 | Reply

2. That’s the neatest thing: the metric $g$ 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.

Comment by John Armstrong | September 21, 2011 | Reply

3. […] now that we’ve introduced the idea of a metric on a manifold, it’s natural to talk about mappings that preserve them. We call such maps […]

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4. […] that we can define the inner product of two vectors using a metric , we want to generalize this to apply to vector […]

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5. […] next step after using a metric to define an inner product on the module of vector spaces over the ring of smooth functions is to […]

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6. […] say that is an orientable Riemannian manifold. We know that this lets us define a (non-degenerate) inner product on differential forms, […]

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7. […] Armstrong: (Pseudo)-Riemannian Metrics, Isometries, Inner Products on 1-Forms, The Hodge Star in Coordinates, The Hodge Star on […]

8. […] want to start getting into a nice, simple, concrete example of the Hodge star. We need an oriented, Riemannian manifold to work with, and for this example we take , which we cover with the usual coordinate […]

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9. […] continue our example considering the special case of as an oriented, Riemannian manifold, with the coordinate -forms forming an oriented, orthonormal basis at each […]

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10. […] some examples will quickly shed some light on this. We can even extend to the pseudo-Riemannian case and pick a coordinate system so that , where . That is, any two are orthogonal, and each either […]

Pingback by The Hodge Star, Squared « The Unapologetic Mathematician | October 18, 2011 | Reply

11. […] why do we care about this particularly? In the presence of a metric, we have an equivalence between -forms and vector fields . And specifically we know that the […]

Pingback by Line Integrals « The Unapologetic Mathematician | October 21, 2011 | Reply

12. how can it help to our daily life ?

Comment by Sheenj Ruizo Ö | October 24, 2011 | Reply

13. […] does this look like when we have a metric and we can rewrite the -form as a vector field ? In this case, is exact if and only if is […]

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14. […] we want another way of viewing this orientation. Given a metric on we can use the inverse of the Hodge star from on the orientation -form of , which gives us a […]

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15. […] a moment, let’s return to the case of Riemannian manifolds; the vector field analogue of an exact -form is called a “conservative” […]

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16. […] -form, not a vector field, but remember that we’re working in our standard with the standard metric, which lets us use the Hodge star to flip a -form into a -form, and a -form into a vector field! […]

Pingback by Gauss’ Law « The Unapologetic Mathematician | January 11, 2012 | Reply

17. И здесь

Comment by charlesse | January 16, 2012 | Reply

18. […] which, though familiar to many, are really heavy-duty equipment. In particular, they rely on the Riemannian structure on . We want to strip this away to find something that works without this assumption, and […]

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19. […] in hand, we need to properly define the Hodge star in our four-dimensional space, and we need a pseudo-Riemannian metric to do this. Before we were just using the standard , but now that we’re lumping in time we […]

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20. 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.

Comment by isomorphismes | August 31, 2014 | Reply