Identities for the curvature tensorNovember 11, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric ${g}$ has to satisfy certain conditions.  (As usual, we denote by $\nabla$ the Levi-Civita connection associated to $g$, and we assume the ground manifold is smooth.)

First of all, we have skew-symmetry

$\displaystyle R(X,Y)Z = -R(Y,X)Z.$

This is immediate from the definition.

Next, we have another variant of skew-symmetry:

Proposition 1 $\displaystyle g( R(X,Y) Z, W) = -g( R(X,Y) W, Z)$  (more…)

The fundamental theorem of Riemannian geometry and the Levi-Civita connectionNovember 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 ${g}$. (more…)

The Riemann curvature tensorNovember 9, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Today I will discuss the Riemann curvature tensor. This is the other main invariant of a connection, along with the torsion. It turns out that on Riemannian manifolds with their canonical connections, this has a nice geometric interpretation that shows that it generalizes the curvature of a surface in space, which was defined and studied by Gauss. When ${R \equiv 0}$, a Riemannian manifold is flat, i.e. locally isometric to Euclidean space.

Rather amusingly, the notion of a tensor hadn’t been formulated when Riemann discovered the curvature tensor.

Given a connection ${\nabla}$ on the manifold ${M}$, define the curvature tensor ${R}$ by

$\displaystyle R(X,Y)Z := \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z.$

There is some checking to be done to show that ${R(X,Y)Z}$ is linear over the ring of smooth functions on ${M}$, but this is a straightforward computation, and since it has already been done in detail here, I will omit the proof.

The main result I want to show today is the following:

Proposition 1

Let ${M}$ be a manifold with a connection ${\nabla}$ whose curvature tensor vanishes. Then if ${s: U \rightarrow M}$ is a surface with ${U \subset \mathbb{R}^2}$ open and ${V}$ a vector field along ${s}$, then$\displaystyle \frac{D}{\partial x} \frac{D}{\partial y} V = \frac{D}{\partial y} \frac{D}{\partial x} V.$ (more…)

Symmetric connections, corrected versionNovember 9, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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My post yesterday on the torsion tensor and symmetry had a serious error.  For some reason I thought that connections can be pulled back.  I am correcting the latter part of that post (where I used that erroneous claim) here. I decided not to repeat the (as far as I know) correct earlier part.

Proposition 1 Let ${s}$ be a surface in ${M}$, and let ${\nabla}$ be a symmetric connection on ${M}$. Then$\displaystyle \frac{D}{\partial x} \frac{\partial}{\partial y} s = \frac{D}{\partial y} \frac{\partial}{\partial x} s.\ \ \ \ \ (1)$  (more…)

The torsion tensor and symmetric connectionsNovember 8, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Today I will discuss the torsion tensor of a Koszul connection. It measures the deviation from being symmetric in a sense defined below.

Torsion

Given a Koszul connection ${\nabla}$ on the smooth manifold ${M}$, define the torsion tensor ${T}$ by

$\displaystyle T(X,Y) := \nabla_X Y - \nabla_Y X - [X,Y].$  (more…)

The tubular neighborhood theoremNovember 5, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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If ${M}$ is a manifold and ${N}$ a compact submanifold, then a tubular neighborhood of ${N}$ consists of an open set ${U \supset N}$ diffeomorphic to a neighborhood of the zero section in some vector bundle ${E}$ over ${N}$, by which $N$ corresponds to the zero section.

Theorem 1 Hypotheses as above, ${N}$ has a tubular neighborhood. (more…)

Geodesics and the exponential mapNovember 4, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space ${T_p(M)}$ at one point to the manifold ${M}$ which is a local isomorphism. This is interesting because it gives a way of saying, “start at point ${p}$ and go five units in the direction of the tangent vector ${v}$,” in a rigorous sense, and will be useful in proofs of things like the tubular neighborhood theorem—which I’ll get to shortly.

Anyway, first I need to talk about geodesics. A geodesic is a curve ${c}$ such that the vector field along ${c=(c_1, \dots, c_n)}$ created by the derivative ${c'}$ is parallel. In local coordinates ${x_1, \dots, x_n}$, here’s what this means. Let the Christoffel symbols be ${\Gamma^k_{ij}}$. Then using the local formula for covariant differentiation along a curve, we get

$\displaystyle D(c')(t) = \sum_j \left( c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) \right) \partial_j,$

so ${c}$ being a geodesic is equivalent to the system of differential equations

$\displaystyle c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) = 0, \ 1 \leq j \leq n.$ (more…)

Covariant derivatives and parallelism for tensorsNovember 3, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Time to continue the story for covariant derivatives and parallelism, and do what I promised yesterday on tensors.

Fix a smooth manifold ${M}$ with a connection ${\nabla}$. Then parallel translation along a curve ${c}$ beginning at ${p}$ and ending at ${q}$ leads to an isomorphism ${\tau_{pq}: T_p(M) \rightarrow T_q(M)}$, which depends smoothly on ${p,q}$. For any ${r,s}$, we get isomorphisms ${\tau^{r,s}_{pq} :T_p(M)^{\otimes r} \otimes T_p(M)^{\vee \otimes s} \rightarrow T_q(M)^{\otimes r} \otimes T_q(M)^{\vee \otimes s} }$ depending smoothly on ${p,q}$. (Of course, given an isomorphism ${f: M \rightarrow N}$ of vector spaces, there is an isomorphism ${M^* \rightarrow N^*}$ sending ${g \rightarrow g \circ f^{-1}}$—the important thing is the inverse.) (more…)

Parallelism determines the connectionNovember 2, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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I’m going to try participating in Charles Siegel’s MaBloWriMo project of writing a short post a day for a month.  In particular, I’m categorizing yesterday’s post that way too.  I’m making no promises about meeting that every day, but much of the material I talk about lends itself to bite-sized pieces anyway.

There is a nice way to tie together (dare I say connect?) the material yesterday on parallelism with the axiomatic scheme for a Koszul connection. In particular, it shows that connections can be recovered from parallelism.

So, let’s pick a nonzero tangent vector ${Y \in T_p(M)}$, where ${M}$ is a smooth manifold endowed with a connection ${\nabla}$, and a vector field ${X}$. Then ${\nabla_Y X \in T_p(M)}$ makes sense from the axiomatic definition. We want to make this look more like a normal derivative.

Now choose a curve ${c: (-1,1) \rightarrow M}$ with ${c(0)=p,c'(0) = Y}$. Then I claim that

$\displaystyle \nabla_Y X = \lim_{s \rightarrow 0} \frac{ \tau_{p, c(s)}^{-1} X(c(s)) - X(p) }{s}.$ (more…)

Covariant derivatives and parallelismNovember 1, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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[Nobody should read this post without reading the excellent comments below.  It turns out that thinking  more generally (via connections on the pullback bundle) clarifies things. Many thanks to the (anonymous) reader who posted them.  -AM, 5/16]

A couple of days back I covered the definition of a (Koszul) connection. Now I will describe how this gives a way to differentiate vector fields along a curve.

Covariant Derivatives

First of all, here is a minor remark I should have made before. Given a connection ${\nabla}$ and a vector field ${Y}$, the operation ${X \rightarrow \nabla_X Y}$ is linear in ${X}$ over smooth functions—thus it is a tensor (of type (1,1)), and the value at a point ${p}$ can be defined if ${X}$ is replaced by a tangent vector at ${p}$. In other words, we get a map ${T(M)_p \times \Gamma(TM) \rightarrow T(M)_p}$, where ${\Gamma(TM)}$ denotes the space of vector fields. We’re going to need this below. (more…)