- #1
cianfa72
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- 306
- TL;DR
- About the definition of congruent curve to a given one along a vector field
Consider the following: suppose there is a smooth vector field ##X## defined on a manifold ##M##.
Take a smooth curve ##\alpha(\tau)## between two different integral curves of ##X## where ##\tau## is a parameter along it. Let ##A## and ##B## the ##\alpha(\tau)## 's intersection points with the 1st and 2nd integral curves respectively.
What is a sensibile notion of curve congruent to ##\alpha(\tau)## along the vector field ##X## starting from the point ##C## on the 1st integral curve ? I believe the answer is the Lie dragging of ##\alpha(\tau)## along ##X## from the starting point ##A## to ##C##.
Is the above correct ? Thanks.
Take a smooth curve ##\alpha(\tau)## between two different integral curves of ##X## where ##\tau## is a parameter along it. Let ##A## and ##B## the ##\alpha(\tau)## 's intersection points with the 1st and 2nd integral curves respectively.
What is a sensibile notion of curve congruent to ##\alpha(\tau)## along the vector field ##X## starting from the point ##C## on the 1st integral curve ? I believe the answer is the Lie dragging of ##\alpha(\tau)## along ##X## from the starting point ##A## to ##C##.
Is the above correct ? Thanks.