Lie group, Riemannian metric, and connection

In summary, the conversation discusses the computation of a Riemannian connection in terms of structure constants and metric. It is mentioned that there is a unique symmetric connection compatible with a Riemannian metric, and reference is given to pages 48-49 of John Milnor's Morse Theory and page 55 of Riemannian Geometry by Do Carmo for the formulas to find the answer.
  • #1
marton
4
0
hello, i have met with a problem. please help me.
A Lie group,with a left-invariant Riemannian metric, i want to compute the connection compatible with the Riemannian metric. C(ij, k) are the structure constants, g(ij) are the metric, then how to compute the Riemannian connection in terms of g(ij) and C(ij, k)?

thanks a lot.
 
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  • #2
there is a unique symmetric connexion compatible with a riemannian metric. see pages 48-49 of john milnors morse theory, for the formulas.
 
  • #3
Also, check out p. 55 in Riemannian Geometry by Do Carmo. The equation in the middle of the page is precisely the way to find the answer to your question.
 

1. What is a Lie group?

A Lie group is a mathematical object that combines the concepts of a group and a smooth manifold. It is a set of elements that can be combined together using a group operation (such as multiplication) and has the additional property that these operations are smooth and continuous. Lie groups are important in many areas of mathematics, physics, and engineering, particularly in the study of symmetry.

2. What is a Riemannian metric?

A Riemannian metric is a mathematical tool used to measure distances and angles on a smooth manifold. It assigns a positive-definite inner product to each point on the manifold, allowing for the calculation of lengths of curves and angles between curves. Riemannian metrics are essential in the study of geometry, especially in the field of differential geometry.

3. How are Lie groups and Riemannian metrics related?

Lie groups and Riemannian metrics are closely connected, as Lie groups are often equipped with a Riemannian metric that is invariant under the group's operations. This means that the metric is preserved when elements of the group are transformed using the group's operations. This relationship is important in the study of geometric structures on Lie groups.

4. What is a connection on a manifold?

A connection on a manifold is a mathematical tool that allows for the comparison of tangent spaces at different points on the manifold. It assigns a linear map to each point on the manifold, providing a way to connect the tangent spaces at those points. Connections are essential in the study of differential geometry, as they allow for the calculation of important geometric quantities such as curvature.

5. How does a connection relate to the geometry of a manifold?

A connection is closely related to the geometry of a manifold, as it provides a way to measure the deviation of a manifold from being flat. The curvature of a manifold can be calculated using the connection, and it is a fundamental quantity in the study of differential geometry. Connections also play a crucial role in the study of Lie groups and Riemannian metrics, as they allow for the comparison of geometric structures on these mathematical objects.

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