Consider a smooth finite-dimensional manifold $M$ with metric $g$ and connection $\nabla$. For some local coordinate system, denote by $g^{\alpha \beta}$ the inverse of the metric tensor and by $\Gamma^\alpha_{\beta\gamma}$ the Christoffel symbols for $\nabla$. The connection $\nabla$ is assumed to be torsion free, but it does in general not equal the Levi-Civita connection for $g$. Denote also by $R^\alpha_{\beta\gamma\eta}$ the curvature tensor built from $\nabla$. We can then define a vector field $h$ by $$ h^\alpha = R^\alpha_{\beta\gamma\eta} g^{\gamma\zeta}\nabla_\zeta g^{\eta\beta}\;. $$ Is there a "hands-on" geometric interpretation for $h$? (Does it maybe even have a name?) In particular, is there a geometric way of "seeing" when $h$ vanishes? (Of course there are the trivial cases where $\nabla$ is flat or Levi-Civita for $g$, but there must be others.)

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    $\begingroup$ So... did you pull that out of your hat or is there a reason you suspect this will have a geometric meaning? $\endgroup$ – Gunnar Þór Magnússon May 21 '15 at 17:28
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    $\begingroup$ @Gunnar Þór Magnússon Good question ;-) It arises naturally when considering a noisy version of the "heat equation" with domain $S^1$ and target manifold $M$. Here, $\nabla$ defines the heat equation and $g$ is determined by the noise. The vector field $h$ is then a logarithmically divergent drift the solution picks up, due to the interaction between the noise term and the heat equation. $\endgroup$ – Martin Hairer May 21 '15 at 18:07
  • $\begingroup$ I have only vague thoughts. The connection defines geodesics. If you start with a frame of tangent vectors at a point, you can study how that frame changes when you parallel transport it along the connection's geodesics. And maybe if you differentiate twice, you get something useful. Your vector field appears to involve a trace of the curvature (so a Ricci-like tensor), so maybe you want to look at how the determinant of the frame changes relative to the volume form of the Riemannian metric. Maybe this vector field measures the direction of maximal rate of change of this? $\endgroup$ – Deane Yang May 23 '15 at 20:34

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