Understanding the Parallel Transport Problem in Riemannian Geometry

In summary, the conversation is about solving a problem related to Riemannian geometry. The problem involves showing that a vector field is parallel if and only if its derivative is perpendicular to a tangent vector. The person is seeking hints and has not yet solved the problem, but plans to use a specific equation to solve it. They also mention having trouble with another problem.
  • #1
drgigi
1
0
Hi!
I've finally decided to tackle a diff geom book, but I'm having trouble with this Problem 4/Chapter 2 from Do Carmo's Riemannian Geometry:

Let [tex]M^2\subset R^3[/tex] be a surface in [tex]R^3[/tex] with induced Riemannian metric. Let [tex]c:I\rightarrow M[/tex] be a differentiable curve on [tex]M[/tex] and let [tex]V[/tex] be a vector field tangent to [tex]M[/tex] along [tex]c[/tex]; [tex]V[/tex] can be thought of as a smooth function [tex]V:I\rightarrow R^3[/tex], with [tex]V(t)\in T_{c(t)}M[/tex].

a)show that [tex]V[/tex] is parallel if and only if [tex]dV/dt[/tex] is perpendicular to [tex]T_{c(t)}\subset R^3[/tex] where [tex]dV/dt[/tex] is the usual derivative of [tex]V:I\rightarrow R^3[/tex]

b) hopefully I can handle myself. will come back if not! :)

So I guess the plan is to use
[tex]DV/dt=(dv^k/dt + \Gamma^k_{ij} v^j dx^i/dt) X_k=0[/tex]
and dot it with some vector [tex]u^iX_i[/tex]. If I can show that the second term in Dv/dt dotted with this u is zero the problem is done, but I don't see why that should be true..
if I dot [tex]X_i[/tex] with [tex]X_j[/tex] i get [tex]\delta_{i,j}[/tex], right? what then?

any hints would be great! Thanks!
 
Physics news on Phys.org
  • #2
And did you solve Problem 3 before getting to Problem 4?
 

What is the Parallel Transport Problem?

The Parallel Transport Problem refers to the challenge of transporting an object or vector along a curved path while maintaining its direction and orientation. This is a fundamental concept in differential geometry and has applications in fields such as physics, engineering, and computer science.

Why is the Parallel Transport Problem important?

The Parallel Transport Problem is important because it allows us to understand how objects move and change in curved spaces. It is also essential for calculating quantities such as curvature and geodesics, which have implications in the study of gravity, electromagnetism, and other physical phenomena.

What are the main techniques used to solve the Parallel Transport Problem?

There are several approaches to solving the Parallel Transport Problem, including using differential equations, geometric methods, and numerical algorithms. Some commonly used techniques include the parallel transport equation, the covariant derivative, and the Levi-Civita connection.

What are some real-world applications of the Parallel Transport Problem?

The Parallel Transport Problem has numerous applications in various fields, including robotics, computer graphics, and navigation systems. It is also used in the study of general relativity, where it helps explain the behavior of particles in curved spacetime.

What are some common misconceptions about the Parallel Transport Problem?

One common misconception about the Parallel Transport Problem is that it only applies to objects moving on the surface of a sphere. In reality, it can be applied to any curved surface or space. Another misconception is that parallel transport always results in a straight path, when in fact, it can lead to curved paths depending on the geometry of the surface.

Similar threads

  • Differential Geometry
Replies
9
Views
5K
  • Differential Geometry
Replies
9
Views
3K
  • Differential Geometry
Replies
5
Views
3K
  • Special and General Relativity
Replies
2
Views
850
  • Differential Geometry
Replies
2
Views
534
  • Differential Geometry
Replies
2
Views
2K
Replies
4
Views
5K
  • Differential Geometry
Replies
2
Views
2K
  • Introductory Physics Homework Help
Replies
17
Views
956
  • Introductory Physics Homework Help
2
Replies
42
Views
3K
Back
Top