Notion of congruent curve along a vector field

Click For Summary
SUMMARY

The discussion centers on the concept of congruent curves within a smooth vector field ##X## defined on a manifold ##M##. The user proposes that the notion of a curve congruent to a smooth curve ##\alpha(\tau)## along the vector field is achieved through the process of Lie dragging from point ##A## on the first integral curve to point ##C##. This method ensures that the curve maintains its properties as it follows the vector field. The user seeks confirmation on the correctness of this approach.

PREREQUISITES
  • Understanding of smooth vector fields
  • Familiarity with integral curves in differential geometry
  • Knowledge of Lie derivatives and Lie dragging
  • Basic concepts of manifolds
NEXT STEPS
  • Study the properties of Lie derivatives in differential geometry
  • Explore the concept of integral curves in smooth vector fields
  • Research the mathematical framework of manifolds and their applications
  • Examine examples of curve congruences in various vector fields
USEFUL FOR

Mathematicians, physicists, and students of differential geometry interested in the behavior of curves in vector fields and the application of Lie dragging in manifold theory.

cianfa72
Messages
2,942
Reaction score
308
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.
 
Physics news on Phys.org

Similar threads

Undergrad About the maximal extension of local charts on a manifold
  • · Replies 8 ·
Replies
8
Views
1K
Undergrad Frobenius theorem applied to frame fields
  • · Replies 30 ·
2
Replies
30
Views
3K
Undergrad Parallel Transport Along Geodesic
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Manifold hypersurface foliation and Frobenius theorem
  • · Replies 73 ·
3
Replies
73
Views
7K
Undergrad Get geodesics & parallel transport on 2D surfaces (discrete timestep)
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Lie dragging vs Fermi-Walker transport along a given vector field
  • · Replies 26 ·
Replies
26
Views
2K
Graduate About lie algebras, vector fields and derivations
  • · Replies 20 ·
Replies
20
Views
4K
Undergrad Congruence of curves in spacetime
  • · Replies 23 ·
Replies
23
Views
2K
Graduate Question about the derivation of the tangent vector on a manifold
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad What are the components of a vector field on a manifold?
  • · Replies 4 ·
Replies
4
Views
3K