Riemannian Metric Tensor & Christoffel Symbols: Learn on R2

[shanky]

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.

 

[Orodruin]

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.

 

[shanky]

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?

 

[Orodruin]

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.

 

[shanky]

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

 

[Orodruin]

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.

 

[shanky]

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

 

[Orodruin]

Your question is still not very clear.