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Let $(M, g)$ be a $2$ dimensional Riemannian manifold.

Then we consider the Riemannian metric on TM described here.

Assume that $X:M\to TM$ is a vector field. For every $p\in M, \quad DX_p(T_pM)$ is a 2 dimensional subspace of the tangent space of $TM$ at point $(p,X(p))$. We define the function $q:M \to \mathbb{R}$ with $$q(p)=\kappa(DX_p(T_pM))$$ where $\kappa $ is the sectional curvature.

To what extend does the function $q$ and its integral $\int_M q\; d\Omega_g$ have some information about the dynamics of $X$? Under what kind of perturbations of $X$, this integral is stable(unchanged)?

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  • $\begingroup$ Is $D$ the Levi-Civita connection? Then isn't $DX_p(T_pM)$ simply a subspace of $T_pM$ (i.e. the covariant derivative of the vector field at $p$ is a tangent vector at $p$)? I also don't see why the subspace should be two-dimensional, e.g. if $X=0$, then the subspace is zero-dimensional. Am I missing something? $\endgroup$
    – S.Surace
    Commented Aug 25, 2018 at 11:37
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    $\begingroup$ $D$ denotes the derivative of the map $X$. The Levi Civita connection is needed to get a Riemannian metric on TM. $\endgroup$
    – Sebastian
    Commented Aug 25, 2018 at 11:46
  • $\begingroup$ @S.Surace D is the derivative as it $\endgroup$
    – Ali Taghavi
    Commented Aug 25, 2018 at 12:48
  • $\begingroup$ as @Sebastian said. $\endgroup$
    – Ali Taghavi
    Commented Aug 25, 2018 at 12:49
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    $\begingroup$ If the vector field is identically zero then q should be the Gaussian curvature, I think $\endgroup$
    – Martin de Borbon
    Commented Aug 25, 2018 at 13:05

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