Parallel Transport: Constancy of Magnitudes & Angles Along Geodesic

In summary, the conversation discusses parallel transport of a vector field Y along a geodesic with tangent vector X in a Riemannian manifold. It is shown that the magnitudes of X and Y, as well as the angle between them, remain constant along the geodesic. The definition of parallel transport, the meaning of "along the geodesic", and the formula for calculating the angle between two vectors in a Riemannian manifold are also mentioned.
  • #1
latentcorpse
1,444
0
A vector field Y is parallely propagated (with respect to the Levi-Civita connection)
along an affinely parameterized geodesic with tangent vector X in a Riemannian
manifold. Show that the magnitudes of the vectors X, Y and the angle between
them are constant along the geodesic.
 
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  • #2
Some questions to get you started.

What is the definition of parallel transport?
What does along the geodesic mean?
How is the angle between two vectors defined?
 
  • #3
betel said:
Some questions to get you started.

What is the definition of parallel transport?
What does along the geodesic mean?
How is the angle between two vectors defined?

The tensor T is parrallely transported along the curve with tangent [itex]X^a[/itex] if [itex]\nabla_X T=0[/itex]

Along the geodesic means along the affinely parameterised curve of shortest distance (think i may be a bit off here but hopefully you can clear it up!)

On a Riemannian manifold, the angle between two vectors is given by

[itex]\theta = \cos^{-1} \left( \frac{ g(X,Y) }{ ( |X||Y| ) } \right)[/itex] where [itex]|X|= \sqrt{ g(X,X)}[/itex]
 
  • #4
So what is your problem?
 
  • #5


Parallel transport is a fundamental concept in differential geometry that plays a crucial role in understanding the behavior of vectors in curved spaces. In the context of Riemannian manifolds, parallel transport is defined as the process of transporting a vector along a path while maintaining its direction and magnitude.

In this scenario, we are considering the parallel transport of a vector field Y along an affinely parameterized geodesic with tangent vector X in a Riemannian manifold. The Levi-Civita connection, which is the unique connection compatible with the metric structure of the manifold, is used to define the parallel transport.

We can show that the magnitudes of the vectors X and Y remain constant along the geodesic by considering the definition of parallel transport. Since the transport is along a geodesic, which is the shortest path between two points on a curved surface, the distance between any two points along the geodesic remains constant. This means that the magnitude of the vector Y, which is being transported along the geodesic, remains the same.

Furthermore, the angle between the vectors X and Y also remains constant along the geodesic. This can be seen by considering the fact that the Levi-Civita connection preserves the inner product between two vectors. Since the inner product is defined in terms of the angle between two vectors, it follows that the angle between X and Y remains constant during parallel transport.

In summary, parallel transport along a geodesic in a Riemannian manifold results in the constancy of magnitudes and angles between vectors. This is a fundamental property of parallel transport that allows us to study the behavior of vectors in curved spaces and is an important concept in many fields of science, including physics and mathematics.
 

1. What is parallel transport?

Parallel transport is a concept in differential geometry that describes the process of moving vectors along a curve or path in a way that preserves their direction and magnitude.

2. What is the significance of parallel transport?

The concept of parallel transport is important in understanding the properties of curved surfaces and spaces. It helps to define geodesics, which are the shortest paths between points on curved surfaces, and also plays a key role in the study of gravity and general relativity.

3. What is the constancy of magnitudes and angles along a geodesic?

The constancy of magnitudes and angles along a geodesic means that when a vector is parallel transported along a geodesic, its magnitude and angle with respect to other vectors will remain unchanged. This is an important property for understanding the behavior of vectors on curved surfaces.

4. How is parallel transport related to curvature?

Parallel transport is closely related to curvature because it is the mechanism by which we define and measure curvature on a curved surface or space. The deviation of a vector from its original direction after being parallel transported along a closed path is a measure of the curvature at that point.

5. Can parallel transport be applied in real-world scenarios?

Yes, parallel transport has numerous applications in fields such as physics, engineering, and computer science. For example, it is used in navigation and mapping systems to calculate the shortest path between two points on a curved surface, and in computer graphics to create realistic 3D models of curved objects.

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