Riemannian Metric Tensor & Christoffel Symbols: Learn on R2

In summary: Can you please elaborate?In summary, the metric tensor defines distances and inner products in any space, while the Christoffel symbols tell you how the vector basis changes with the point in space.
  • #1
shanky
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Hi,

Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.
 
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  • #2
The metric tensor defines distances and inner products regardless of which space you are looking at. The Christoffel symbols tell you how your vector basis changes with the point in space. The exact expressions for the metric and Christoffel symbols depend on the chosen coordinate system.
 
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Likes shanky
  • #3
Orodruin said:
The metric tensor defines distances and inner products regardless of which space you are looking at. The Christoffel symbols tell you how your vector basis changes with the point in space. The exact expressions for the metric and Christoffel symbols depend on the chosen coordinate system.

Thanks Orodruin.
But I want metric tensor(MT) and Christoffel symbols(CS) in R2 perspective? Consider any 3D object in space how does MT and CS look in R2?
What do you mean by choosing coordinate system ? Is it world / Cartesian coordinate system you mean?
 
  • #4
It is unclear what you mean by "a 3D object in space ... Look in R2".

In order to express the components of the metric and the Christoffel symbols you need to select the coordinate system in which you want to find the components.
 
  • #5
Orodruin said:
It is unclear what you mean by "a 3D object in space ... Look in R2".

In order to express the components of the metric and the Christoffel symbols you need to select the coordinate system in which you want to find the components.

3D object in Riemannian Space ... real vector space with D=2 dimensions
 
  • #6
Your use of the term "3D object" is confusing. What you are likely referring to is a two-dimensional manifold embedded in three dimensions.

A manifold in general is not a vector space - its tangent spaces are.
 
  • #7
Orodruin said:
Your use of the term "3D object" is confusing. What you are likely referring to is a two-dimensional manifold embedded in three dimensions.

A manifold in general is not a vector space - its tangent spaces are.

Yah its manifold - tangent spaces. My main question was what does Riemannian metric tensor and Christoffel symbols on R2 mean? Illustrate with example
 
  • #8
Your question is still not very clear.
 

1. What is a Riemannian metric tensor?

A Riemannian metric tensor is a mathematical object that describes the geometry of a curved surface or space. It assigns a metric, or distance, to each point on the surface, allowing for the measurement of distances, angles, and other geometric properties.

2. What is the significance of the Christoffel symbols?

The Christoffel symbols are used to define the connection between the metric tensor and the curvature of a space. They represent how the metric tensor changes as one moves along different directions on a curved surface, and are essential for understanding the geometry of Riemannian spaces.

3. How are Riemannian metric tensors and Christoffel symbols related?

The Christoffel symbols are derived from the components of the Riemannian metric tensor. They are used to define the connection between the metric tensor and the curvature of a space, and are crucial for understanding the geometry of Riemannian spaces.

4. What is the difference between R2 and Riemannian spaces?

R2, or Euclidean space, is a flat, two-dimensional space with a constant curvature. Riemannian spaces, on the other hand, can have varying curvatures and are not limited to two dimensions. They are used to describe curved surfaces and spaces, and the concepts of R2 do not apply to them.

5. How can I use Riemannian metric tensor and Christoffel symbols in my research?

Riemannian geometry is used in many fields, including physics, engineering, and computer science. The concepts of Riemannian metric tensors and Christoffel symbols can be applied to study and model complex systems, such as fluid dynamics, general relativity, and computer vision. They can also be used for optimization problems and machine learning algorithms.

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