Let $M$ be an $n$-dimensional $C^{\infty}$ manifold. Two vector fields $X$ and $Y$ on $M$ that never vanish is said to have the same direction if their tangent vectors $X_p$ and $Y_p$ have the same direction for each $p\in M$. Can anyone kindly clarify what is meant by the statement "their tangent vectors have the same direction" ?.

Thanks in advance.

  • 3
    $\begingroup$ Two vectors have the same direction if one is a positive scalar multiple of the other. Thus, the way I understand the statement, is that there exists a positive-valued function $f(p)$ such that $X_{p} = f(p) Y_{p}$ for each $p \in M$. $\endgroup$
    – avs
    Jun 14, 2017 at 19:19
  • $\begingroup$ Excuse me if this seemed stupid. Does saying that "Two tangent vectors have an opposite direction" mean that one is a "negative" scalar of the other ? @avs $\endgroup$ Jun 14, 2017 at 19:32
  • 1
    $\begingroup$ Seems reasonable to me, not stupid. $\endgroup$
    – avs
    Jun 14, 2017 at 19:37

1 Answer 1


If you want to visualize the situation you have to think to a 2D manifold in a 3D space ( all we have some difficult to visualize more than 3 dimensions). In this case two vector spaces have te same direction if the vectors stay on the same tangent line at any point of the manifold ( you can visualize two vectors at a point of a sphere for the simpler example). This mens that the two vectors have the same direction but can have different, and also opposite, lenght.


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