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The Hahn-Banach theorem and two applications November 28, 2009

Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
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I have been finishing my MaBloWriMo series on differential geometry with a proof of the Myers comparison theorem, which right now has only an outline, but will rely on the second variation formula for the energy integral.  After that, it looks like I’ll be posting somewhat more randomly.   Here I will try something different.

The Hahn-Banach theorem is a basic result in functional analysis, which simply states that one can extend a linear function from a subspace while preserving certain bounds, but whose applications are quite manifold.

Edit (12/5): This material doesn’t look so great on WordPress.  So, here’s the PDF version.  Note that the figure is omitted in the file.

The Hahn-Banach theorem             

Theorem 1 (Hahn-Banach) Let {X} be a vector space, {g: X \rightarrow \mathbb{R}_{\geq 0}} a positive homogeneous (i.e. {g(tx) = tg(x), t >0}) and sublinear (i.e. {g(x+y) \leq g(x) + g(y)}) function. 

Suppose {Y} is a subspace and {\lambda: Y \rightarrow \mathbb{R}} is a linear function with {\lambda(y) \leq g(y)} for all {y \in Y}.

Then there is an extension of {\lambda} to a functional {\tilde{\lambda}: X \rightarrow \mathbb{R}} with

\displaystyle \tilde{\lambda}(x) \leq g(x), \ x \in X. 
 
 
I’ll omit the proof; I want to discuss why it is so interesting.  One of its applications lies in questions of the form “are elements of this form dense in the space”? The reason is that if {X} is a normed linear space and {Y} a closed subspace, the quotient vector space {X/Y} is a norm with the norm {|x+Y| := \inf_{y\in Y} |x-y|.} (The closedness condition is necessary because otherwise there might be nonzero elements of {X/Y} with zero norm.) (more…)

A theorem of Mazur-Ulam on isometric maps of vector spaces November 22, 2009

Posted by Akhil Mathew in analysis, functional analysis, MaBloWriMo.
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I first posted this entry at Climbing Mount Bourbaki, where I have continued the MaBloWriMo series into topics in Riemannian geometry such as the Cartan-Hadamard theorem.  This particular material came up as part of the proof that distance-preserving maps between Riemannian manifolds are actually isometries.  However, the style of the entry seemed appropriate for this blog, so I’m placing it here as well.

The result in question is:

Theorem 1 (Mazur-Ulam) An isometry {M: X \rightarrow X'} of a normed linear space {X} onto another normed linear space {X'} with {M(0)=0} is linear. (more…)

Hopf-Rinow II and an application November 15, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifold {(M,g)} which is a metric space {d}, the existence of arbitrary geodesics from {p} implies that {M} is complete with respect to {d}. Actually, this is slightly stronger than what H-R states: geodesic completeness at one point {p} implies completeness.

The first thing to notice is that {\exp: T_p(M) \rightarrow M} is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from {p} exist. Moreover, it is surjective by the second Hopf-Rinow theorem.

Now fix a {d}-Cauchy sequence {q_n \in M}. We will show that it converges. Draw minimal geodesics {\gamma_n} travelling at unit speed with

\displaystyle \gamma_n(0)=p, \quad \gamma_n( d(p,q_n)) = q_n.  (more…)

The Hopf-Rinow theorems and geodesic completeness November 14, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Ok, yesterday I covered the basic fact that given a Riemannian manifold {(M,g)}, the geodesics on {M} (with respect to the Levi-Civita connection) locally minimize length. Today I will talk about the phenomenon of “geodesic completeness.”

Henceforth, all manifolds are assumed connected.

The first basic remark to make is the following. If {c: I \rightarrow M} is a piecewise {C^1}-path between {p,q} and has the smallest length among piecewise {C^1} paths, then {c} is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick {a,b \in I} very close to each other, so that {c([a,b])} is contained in a neighborhood of {c\left( \frac{a+b}{2}\right)} satisfying the conditions of yesterday’s theorem; then {c|_{[a,b]}} must be length-minimizing, so it is a geodesic. We thus see that {c} is locally a geodesic, hence globally.

Say that {M} is geodesically complete if {\exp} can be defined on all of {TM}; in other words, a geodesic {\gamma} can be continued to {(-\infty,\infty)}. The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)

The following are equivalent:

  • {M} is geodesically complete.
  • In the metric {d} on {M} induced by {g} (see here), {M} is a complete metric space (more…)

Geodesics are locally length-minimizing November 13, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Fix a Riemannian manifold with metric {g} and Levi-Civita connection {\nabla}. Then we can talk about geodesics on {M} with respect to {\nabla}. We can also talk about the length of a piecewise smooth curve {c: I \rightarrow M} as

\displaystyle l(c) := \int g(c'(t),c'(t))^{1/2} dt .

 Our main goal today is:

Theorem 1 Given {p \in M}, there is a neighborhood {U} containing {p} such that geodesics from {p} to every point of {U} exist and also such that given a path {c} inside {U} from {p} to {q}, we have

 

\displaystyle l(\gamma_{pq}) \leq l(c)  

with equality holding if and only if {c} is a reparametrization of {\gamma_{pq}}.

In other words, geodesics are locally path-minimizing.   Not necessarily globally–a great circle is a geodesic on a sphere with the Riemannian metric coming from the embedding in \mathbb{R}^3, but it need not be the shortest path between two points. (more…)

The test case: flat Riemannian manifolds November 12, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Recall that two Riemannian manifolds {M,N} are isometric if there exists a diffeomorphism {f: M \rightarrow N} that preserves the metric on the tangent spaces. The curvature tensor  (associated to the Levi-Civita connection) measures the deviation from flatness, where a manifold is flat if it is locally isometric to a neighborhood of {\mathbb{R}^n}.

Theorem 1 (The Test Case) The Riemannian manifold {M} is locally isometric to {\mathbb{R}^n} if and only if the curvature tensor vanishes. (more…)

Identities for the curvature tensor November 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 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}. (more…)

The Riemann curvature tensor November 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 version November 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…)

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