[Math] Riemannian Geometry

dg.differential-geometryriemannian-geometry

I come from a background of having done undergraduate and graduate courses in General Relativity and elementary course in riemannian geometry.

Jurgen Jost's book does give somewhat of an argument for the the statements below but I would like to know if there is a reference where the following two things are proven explicitly,

  1. That the sectional curvature of a 2-dimensional subspace of a tangent space at a point on the Riemannian manifold is independent of the choice of basis. That is the definition of the sectional curvature depends only on the choice of the 2-dim subspace.

  2. That the sectional curvature determines the Riemannian curvature fully.

Secondly can one give me a reference where I can see how in practice is sectional curvature computed. To a first timer to this subject it is not obvious how one does a calculation on "all" 2-dimensional subspaces of a high-dimensional space. Especially when people talk of manifolds with "constant sectional curvature". How are they realized?

I would like to see some explicit examples to understand this point.

Further some studies about homogeneous spaces (needed to understand some issues in Quantum Field Theory) got me to the following 4 very non-trivial ideas in Riemannian manifolds which I am stating in my own way here ,

  1. That the isometry group of a Riemannian manifold is always a lie group.

  2. The isotropy subgroup of any point on a Riemannian manifold under the
    smooth transitive action of its own isometry group on itself is a compact
    subgroup. (The context being what is called a "Riemannian Homogeneous Space")

  3. {This point was earlier framed in a way which made the bi-implication false as pointed out by some people} The formulation should be as follows.

    A Riemannian Homogeneous Space is a riemannian manifold on which the isometry group acts transitively. Now the theorem is that such a space is compact IFF its isometry group is compact.

    Thats the statement whose intuition I am looking for.

    Apologies for the confusion caused.

  4. { This question too was not framed properly. Basically I could not figure out how to write the nabla for the connection! It should be as Jose has pointed out.}

    A riemannian manifold is locally symmetric if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection.

Can one give me the intuition behind these or give me specific references where these are proven in explicit details?

Best Answer

To get a better feel of the Riemann curvature tensor and sectional curvature:

  1. Work through one of the definitions of the Riemann curvature tensor and sectional curvature with a $2$-dimensional sphere of radius $r$.
  2. Define the hyperbolic plane as the space-like "unit sphere" of $3$-dimensional Minkowski space, defined using an inner product with signature $(-,+,+)$. Work out the sectional and Riemann curvature of that
  3. Repeat #1 and #2 for the $n$-dimensional sphere and hyperbolic space, as well as flat space

Sectional curvature determines Riemann curvature:

That the sectional curvature uniquely determines the Riemann curvature is a consequence of the following:

  1. The Riemann curvature tensor is a quadratic form on the vector space of $\Lambda^2T_xM$
  2. The sectional curvature function corresponds to evaluating the Riemann curvature tensor (as a quadratic form) on decomposable elements of $\Lambda^2T_xM$
  3. There is a basis of $\Lambda^2T_xM$ consisting only of decomposable elements

Added in response to Anirbit's comment

Perhaps you shouldn't try to compute the curvature too soon. First, make sure you understand the Riemannian metric of the unit sphere and hyperbolic space inside out. There are many ways to do this. But the most concrete way I know is to use stereographic projection of the sphere onto a hyperplane orthogonal to the last co-ordinate axis. Either the hyperplane through the origin or the one through the south pole works fine. This gives you a very nice set of co-ordinates on the whole sphere minus one point. Work out the Riemannian metric and the Christoffel symbols. Also, work out formulas for an orthonormal frame of vector fields and the corresponding dual frame of 1-forms. Figure out the covariant derivatives of these vector fields and the corresponding dual connection 1-forms.

After you do this, do everything again with hyperbolic space, which is the hypersurface

$-x_0^2 + x_1^2 + \cdots + x_n^2 = -1$ with $x_0 > 0$

in Minkowski space with the Riemannian metric induced by the flat Minkowski metric. You can do stereographic projection just like for the sphere but onto the unit $n$-disk given by

$x_1^2 + \cdots + x_n^2 = 1$ and $x_0 = 0$,

where the formula for the hyperbolic metric looks just like the spherical metric in stereographic co-ordinates but with a sign change in appropriate places. This is the standard conformal model of hyperbolic space.

After you understand this inside out, you can use these pictures to figure out why the $n$-sphere and its metric is given by $O(n+1)/O(n)$ and hyperbolic space by $O(n,1)/O(n)$ and why the metrics you've computed above correspond to the natural invariant metric on these homogeneous spaces. You can then check that the formulas for invariant metrics on homogeneous spaces give you the same answers as above.

Use references only for the general formulas for the metric, connection (including Christoffel symbols), and curvature. I recommend that you try to work out these examples by hand yourself instead of trying to follow someone else's calculations. If possible, however, do it with another student who is also trying to learn this at the same time.

If, however, you want to peek at a reference for hints, I recommend the book by Gallot, Hulin, and Lafontaine. I suspect that the book by Thurston is good too (I studied his notes when I was a student). For invariant Riemannian metrics on a homogeneous space, I recommend the book by Cheeger and Ebin (available cheap from AMS! When I was a student, I had to pay a hundred dollars for this little book but it was well worth it).

But mostly when I was learning this stuff, I did and redid the same calculations many times on my own. I was never able to learn much more than a bare outline of the ideas from either books or lectures. Just try to get a rough idea of what's going on from the books, but do the details yourself.

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