Coordinates in Riemannian Geometry

In summary, coordinates in Riemannian Geometry are a set of numbers used to specify the position of a point in a Riemannian manifold. They are related to curvature, can be visualized, and there are different types and ways to transform them.
  • #1
KalyanK
1
0
Hi,

I was wondering if Geodesic polar coordinates, Geodesic shperical coordinates and Riemann Normal coordinates are the same. Also, are there any standard techniques for computing these coordinates for a manifold given in terms of level set of a function. Are there any good references that deal with these topics? Thanks.

Kalyan
 
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  • #2
Yes, they are the same.
The good source is, probably, W. Klingenberg, "Riemannian geometry".
 
  • #3


Hi Kalyan,

Geodesic polar coordinates, geodesic spherical coordinates, and Riemann normal coordinates are not exactly the same, but they are all related concepts in Riemannian geometry. Let's break down each one and discuss their differences.

Geodesic polar coordinates refer to a coordinate system on a Riemannian manifold where the coordinate lines are geodesics. This means that the coordinate lines follow the shortest path between two points on the manifold. These coordinates are useful for studying geodesic equations and properties of curves on the manifold.

Geodesic spherical coordinates are a specific type of polar coordinate system on a Riemannian manifold. They are similar to geodesic polar coordinates, but they are specifically used for studying properties of spheres on the manifold. In this coordinate system, the coordinate lines are geodesics that intersect at a common point, similar to how longitude and latitude lines intersect at the North or South pole on a globe.

Riemann normal coordinates, also known as normal coordinates, are a type of local coordinate system on a Riemannian manifold. They are defined at a specific point on the manifold and are based on the tangent space at that point. In these coordinates, the coordinate lines are geodesic segments that start at the chosen point and follow the direction of the tangent vectors at that point. These coordinates are useful for studying properties of the manifold near a specific point.

As for computing these coordinates for a manifold given in terms of level sets of a function, there are various techniques and algorithms that can be used. One common approach is to use a numerical method, such as the Newton-Raphson method, to solve the geodesic equations and find the coordinates at each point on the manifold. Another approach is to use a transformation to map the coordinates from a known manifold to the desired manifold. There are also software packages and libraries that can assist in computing these coordinates.

For references on these topics, I would recommend looking into textbooks on Riemannian geometry or differential geometry. Some good ones include "Riemannian Geometry" by Manfredo do Carmo and "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo. There are also many online resources and lectures available on these topics. I hope this helps clarify the differences between these coordinates and provides some guidance on how to compute them. Good luck with your studies!

 

1. What are coordinates in Riemannian Geometry?

Coordinates in Riemannian Geometry are a set of numbers used to specify the position of a point in a Riemannian manifold. They are analogous to latitude and longitude in traditional geometry, but in Riemannian Geometry, they are not necessarily based on a flat grid system.

2. How are coordinates related to curvature in Riemannian Geometry?

The curvature of a Riemannian manifold can be described in terms of the behavior of coordinates on that manifold. For example, if the coordinates in a certain direction are changing more rapidly than in another direction, this indicates positive curvature in that direction.

3. Can coordinates in Riemannian Geometry be visualized?

Yes, coordinates in Riemannian Geometry can be visualized using various methods such as coordinate charts, embedding diagrams, and geodesics. These visualizations can help us understand the behavior of coordinates on a curved manifold.

4. Are there different types of coordinates in Riemannian Geometry?

Yes, there are several types of coordinates used in Riemannian Geometry, including local coordinates, isothermal coordinates, and geodesic normal coordinates. Each type has its own advantages and is useful for different purposes.

5. How are coordinates transformed in Riemannian Geometry?

In Riemannian Geometry, coordinates can be transformed using a transformation called a coordinate transformation or a change of coordinates. This transformation preserves the underlying geometry of the manifold, and allows us to work with different coordinate systems for different purposes.

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