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The fundamental theorem of Riemannian geometry and the Levi-Civita connection November 10, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Ok, now onto the Levi-Civita connection. Fix a manifold ${M}$ with the pseudo-metric ${g}$. This means essentially a metric, except that ${g}$ as a bilinear form on the tangent spaces is still symmetric and nondegenerate but not necessarily positive definite. It is still possible to say that a pseudo-metric is compatible with a given connection.

This is the fundamental theorem of Riemannian geometry:

Theorem 1 There is a unique symmetric connection ${\nabla}$ on ${M}$ compatible with ${g}$.

The connection is called the Levi-Civita connection, even though Christoffel apparently discovered it first.

Recall that ${g}$ is compatible with ${\nabla}$ if and only if ${\nabla_X g = 0}$ for all ${X}$. Since

$\displaystyle X(g(X_1,X_2)) = (\nabla_X g)(X_1,X_2) + g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2) ,$

by the fact that covariant differentiation commutes with contractions and satisfies the derviative identity, compatibility is equivalent to

$\displaystyle X (g (X_1,X_2)) = g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2),\ \ \ \ \ (1)$

for all ${X,X_1, X_2}$.

Now assume ${M \subset \mathbb{R}^n}$ and we have such a connection associated to ${g}$. Then the connection is uniquely determined by a bunch of Christoffel symbols, which we will determine in terms of ${g}$ by a bit of elementary algebra. In other words, we just need to compute ${\nabla_{\partial_i} \partial_j}$. Now

$\displaystyle \partial_k g( \partial_i, \partial_j) = g( \nabla_{\partial_k} \partial_i, \partial_j) + g( \partial_i, \nabla_{\partial_k}\partial_j).$

We can get two other equations by cyclic permutation:

$\displaystyle \partial_i g( \partial_j, \partial_k) = g( \nabla_{\partial_i} \partial_j, \partial_k) + g( \partial_j, \nabla_{\partial_i}\partial_k)$

$\displaystyle \partial_j g( \partial_k, \partial_i) = g( \nabla_{\partial_j} \partial_k, \partial_i) + g( \partial_k, \nabla_{\partial_j}\partial_i)$

So let ${S_{ij} := \nabla_{\partial_i} \partial_j = \nabla_{\partial_j} \partial_i}$, by symmetry. Let ${T_{ijk} := \partial_i g( \partial_j, \partial_k)}$; these are smooth real functions. These equations can be written

$\displaystyle T_{kij} = g( S_{ik}, \partial_j) + g( S_{jk}, \partial_i)$

$\displaystyle T_{ijk} = g( S_{ij}, \partial_k) + g( S_{ik}, \partial_j)$

$\displaystyle T_{jki} = g( S_{jk}, \partial_i) + g( S_{ij}, \partial_k)$

These are three linear equations in the unknowns ${g( S_{ik}, \partial_j), g( S_{jk}, \partial_i), g( S_{ij}, \partial_k)}$. The system is nonsingular, so we get a unique solution, and consequently by nondegeneracy a unique possibility for the ${S_{ij}}$.

We have just shown the uniqueness assertion of the theorem, which is local. Connections restrict to connections on open subsets.

Now for existence. Still with the hypothesis on ${M}$, choose ${S_{ij}}$ to satisfy the system of three equations outlined above where ${i. Then set ${S_{ji} := S_{ij}}$, and we have a connection ${\nabla}$ with ${\nabla_{\partial_i} \partial_j := S_{ij}}$ since the vector fields ${\partial_i}$ are a frame (i.e. a basis at each tangent space on ${M}$). It is symmetric, since the torsion ${T}$ vanishes (by ${S_{ij}=S_{ji}}$) on pairs ${(\partial_i,\partial_j)}$, and hence identically—since it is a tensor. Now to check for compatibility.

The difference of the two terms in (1) vanishes when ${X,X_1,X_2}$ are of the form ${\partial_i}$. The vanishing holds generally because the difference of the two sides, which is ${(\nabla_X g)(X_1,X_2)}$, is a tensor. Hence compatibility follows.

So we have proved the theorem when ${M}$ is an open submanifold of ${\mathbb{R}^n}$ (though not necessarily with the canonical metric ${\sum_{i=1}^n dx_i \otimes dx_i}$). In general, cover ${M}$ by open subsets ${U_i}$ diffeomorphic to an open set in ${\mathbb{R}^n}$. We get connections ${\nabla_i}$ on ${U_i}$ compatible with ${g|_{U_i}}$.

I claim that ${\nabla_i|_{U_i \cap U_j} = \nabla_j|_{U_i \cap U_j}}$. This is an easy corollary of uniquness. So we can patch the connections together to get the one Levi-Civita connection on ${M}$. In more detail, the whole idea of patching works as follows. Given ${X,Y}$ on ${M}$ and ${p \in M}$, choose a neighorhood ${U}$ containing ${p}$ but contained in some ${U_i}$. Multiply ${X,Y}$ by cutoff functions which are 1 close to ${p}$ but zero outside ${U}$ to get ${X',Y'}$; then set ${\nabla_X Y := \nabla_{i, X'}Y'}$.

Comments»

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2. Ngô Quốc Anh - December 5, 2009

Aha, I think for the existence part, you can simplify as following: You have a metric g then you can define $g_{ij}$, next you can construct $g^{ij}$ then we have Christoffel symbols $\Gamma_{ij}^k$. So that we can define a connection $\nabla : TM \times TM \to TM$ as following

$\displaystyle\nabla_{\frac{\partial}{\partial x^i}}\frac{\partial}{\partial x^j}=\Gamma_{ij}^k\frac{\partial}{\partial x^k}$.

Now we just verify that $\nabla$ is indeed a connection.

Akhil Mathew - December 5, 2009

Not sure how this is a simplification–after all, I described above how to compute the Christoffel symbols (just used $S_{ij}$ instead); locally, the Christoffel symbols can be arbitrarily prescribed for a connection.

3. Ngô Quốc Anh - December 5, 2009

For the uniqueness part, am I right that you have not proposed its proof? What we need is just the so-called “Vanishing Vector Fields” lemma.

Akhil Mathew - December 5, 2009

Uniqueness was checked locally because the Christoffel symbols were uniquely determined by the linear equations above. Hence, it’s true globally. What’s the vanishing vector fields lemma?

Ngô Quốc Anh - December 5, 2009

Any vector field that vanishes at $p$ can be written as a linear combination of other vector fields with coefficients that vanish at $p$.

Akhil Mathew - December 5, 2009

To go from local to global uniqueness, you just need to multiply by a cutoff function. For instance, a connection on a manifold induces a connection on an open submanifold in a unique manner.

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