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Let $M$ be a smooth manifold. I am trying to construct a (piecewise smooth) vector field $V$ along a curve $c$ that takes on prescribed values $K_{i}$ at times $t_{i}$. Say we construct V along $c|_{[t_{i},t_{i+1}]}$. I want to do the following:

Let $\tilde{K_{i}}$ and $\tilde{K_{i+1}}$ be the parallel translations of $K_{i}\in T_{c(t_{i})}M$ and $K_{i+1}\in T_{c(t_{i+1})}M$ along $c|_{[t_{i},t_{i+1}]}$. I define
$V(t)=(\frac{t}{t_{i}-t_{i+1}}+\frac{t_{i+1}}{t_{i+1}-t_{i}})\tilde{K_{i}}|_{c(t)} + (\frac{t}{t_{i+1}-t_{i}}-\frac{t_{i}}{t_{i+1}-t_{i}})\tilde{K_{i+1}}|_{c(t)}$ on $[t_{i},t_{i+1}]$. Does this seem correct?

studiosus
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There are many ways, and it depends on what other properties you wish for the vector. One way is to take small balls around $t_i$'s so that they are disjoint. On each ball $K_i$ can be extended to a vector field on that ball. Then on each ball have a smooth function with support inside these balls and $1$ at $K_i$. Now multiply your vector field with the corresponding function on each ball. Your final $V$ will be defined to equal these on the balls, and $0$ outside the balls.

Behnam Esmayli
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  • @Benham. How exactly we can extend $K_i$ to a vector field in the ball ? Is it just let the vector field to have constant coefficient equal to $K_i$ ? – Kelvin Lois Sep 05 '17 at 22:53
  • Yes, locally you are in $R^n$. So, you can take the constant field. – Behnam Esmayli Sep 07 '17 at 13:07
  • I already made such construction. Can you take a look at this one ? https://math.stackexchange.com/questions/2419330/using-bump-function-in-coordinate-chart-to-construct-a-vector-field-v-along-a – Kelvin Lois Sep 08 '17 at 03:49
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    I have the feeling that your problem is with being convinced than math rigor. If so, I would recommend writing down calculations for one specific example, like on Euclidean plane. It should allay your doubts. – Behnam Esmayli Sep 08 '17 at 19:10
  • Thanks for the advice. You say that there is many ways to construct such vector field. Is there any reference where i can look up one of them ? I'm using john lee's book by the way. Thanks again. – Kelvin Lois Sep 09 '17 at 13:30
  • By many ways, I mean you can have those local extensions in different ways, other than the constant vector field. So you get different extensions. – Behnam Esmayli Sep 09 '17 at 15:10
  • All right. Thank you very much @Behnam. – Kelvin Lois Sep 09 '17 at 15:16