User mathieu anel - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:26:02Z http://mathoverflow.net/feeds/user/27735 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20493/what-is-torsion-in-differential-geometry-intuitively/111198#111198 Answer by Mathieu ANEL for What is torsion in differential geometry intuitively? Mathieu ANEL 2012-11-01T19:34:45Z 2012-11-01T19:34:45Z <p>Torsion is easy to understand but this knowledge seems to be lost. I had to go back to Elie Cartan's articles to find an intuitive explanation (for example, chapter 2 of <a href="http://www.numdam.org/numdam-bin/fitem?id=ASENS_1923_3_40__325_0" rel="nofollow">http://www.numdam.org/numdam-bin/fitem?id=ASENS_1923_3_40__325_0</a>).</p> <p>Let $M$ be a manifold with a connection on its tangent bundle. The basic idea is that any path $\gamma$ in $M$ starting at $x\in M$ can be lifted as a path $\tilde\gamma$ in $T_xM$, but is the $\gamma$ is a loop $\tilde \gamma$ need not be a loop. The resulting translation of the end point <em>is</em> the torsion (or its macroscopic version).</p> <p>The situation is easy in a Lie group $G$ (which I imagine Cartan had in mind). $G$ has a canonical flat connection for which the parallel vectors fields are left invariant vectors fields. For this connection the parallel transport is simply the left translation. The Maurer-Cartan form $\alpha$ is then the parallel transport to the tangent space $T_1G$ at the identity $1\in G$.</p> <p>If $\gamma:[0,1]\to G$ is a path in $G$ starting at $1$. $\gamma'$ is a path in $TG$ and $\alpha(\gamma')$ is a path in $T_1M$. $\alpha(\gamma')$ can be integrated to another path $\tilde \gamma$ in $T_1M$. Let $\gamma_{\leq x}$ be the path $\gamma:[0,x]\to G$, then we define $$\tilde \gamma(x) = \int_0^x\alpha(\gamma'(t))dt = \int_{\gamma_{\leq x}}\alpha.$$ In the sense given by the connection, $\gamma$ and $\tilde\gamma$ have the same speed and the same starting point, so they are the same path (but in different spaces).</p> <p>If $\gamma$ is a loop and $D$ a disk bounding $\gamma$, $\tilde\gamma$ is a loop iff $\tilde\gamma(1)=0\in T_1G$. We have $$\tilde\gamma(1) = \int_\gamma\alpha = \int_Dd\alpha.$$ $\tilde\gamma$ is a loop iff this integral is zero.</p> <p>Now, $\alpha$ can be viewed as the solder form for $TG$, so the torsion is the covariant differential $T=d^\nabla\alpha$. As the connection is flat $T$ reduces to $T=d\alpha$. The Maurer-Cartan equation gives an explicit formula: $T=d\alpha = -\frac{1}{2}[\alpha,\alpha]$. The previous integral is then the integral of the torsion $$\tilde\gamma(1) = \int_Dd\alpha = -\frac{1}{2}\int_D[\alpha,\alpha]$$ and may not be zero.</p> <p>The situation is the same for a general manifold, but the parallel transport is not explicit and formulas are harder.</p> <p>The notion behing this is that of <em>affine connection</em>. As I understand it, an affine connection is a data that authorize to picture the geometry of $M$ inside the tangent space $T_xM$ of some point $x$. If I move away from $x$ in $M$, there will be a corresponding movement away from the origin in $T_xM$ (this is the above lifting of path). If I transport in parallel a frame with me, the frame will move in $T_xM$. Globally the movement of my point and frame is encoded by a family of affine transformations in $T_xM$.</p> <p>Of course this picture of the geometry of $M$ in $T_xM$ is not faithful. Because of the torsion, if I have two paths in $G$ starting at $x$ and ending at the same point, they may not end at the same point in $T_xM$. Because of curvature, even if my two lifts end at the same point, my two frames may not be parallel. The picture is faithful if $M$ is an affine space iff both torsion and curvature vanish (Cartan's structural equations for affine space).</p> <p>I think torsion is beautiful :)</p>