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In DoCarmo’s Riemannian Geometry book, an affine connection $\nabla$ on a smooth manifold $M$ is defined to be a mapping :$\Xi(M)\times \Xi(M)\rightarrow \Xi(M)$, where $\Xi(M)$ is the set of all smooth vector fields on $M$.

Now DoCarmo wants to clarify this definition with a theorem containing the following fact: ... c) if $V$ is a vector field along the curve $c$, and if $V$ is induced by a vector field $Y\in \Xi(M)$, then the covariant derivative of $V$ along $c$ denoted by $\frac{DV}{dt}$ equals $\nabla(\frac{dc}{dt},Y)$, where $\frac{dc}{dt}$ is the velocity field of the curve $c$.

But the domain of $\nabla$ does not include $(\frac{dc}{dt},Y)$, since $\frac{dc}{dt}$ does not belong to $\Xi(M)$. What is the justification?

  • You can extend $\frac{dc}{dt}$ to a smooth vector field on the manifold, and the value of the directional derivative along $c$ is independent of that choice. – Charlie Frohman Nov 23 '18 at 13:05
  • DoCarmo immediately after defining $\frac{dc}{dt}$ states that vector fields along a curve do not necessarily extend to the whole manifold @CharlieFrohman –  Nov 23 '18 at 13:09
  • https://math.stackexchange.com/questions/1865661/vector-fields-on-a-manifold-and-terminology. According to this, the velocity field is generally not a vector field on the whole manifold –  Nov 23 '18 at 13:14
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    The connection is defined globally on sections of the tangent bundle. Then one can show that the value of the covariant derivative $\nabla_XY$ at a point $p$ actually only depends on the value $X_p$ and $Y$ in a neighborhood of $p$. This allows for the $\frac{D}{dt}$ to be well-defined. – Matt Nov 23 '18 at 14:02
  • Thanks, I got it completely now @CharlieFrohman and Matt –  Nov 23 '18 at 15:17
  • Just to clarify, the velocity field is NOT always extendible to a vector field, even locally. For example, take a self-intersecting smooth curve. Other kind of obstructions arise to the curve being an integral curve if we allow the curve to "almost" intersect itself, as in the figure eight example. – Laz Nov 01 '21 at 04:26

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