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zhangzujin
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Let G be a Lie group. Show that there exists a unique affine connection such that [tex]\nabla X=0[/tex] for all left invariant vector fields. Show that this connection is torsion free iff the Lie algebra is Abelian.
Aha. Of course not. I'm just reading Riemannian Geometry by Petersen, interested in the exercises of that.hamster143 said:Homework?
A Lie group is a type of mathematical group that is continuous and smooth. It is used to study symmetries and transformations in various areas of mathematics, such as geometry and physics.
Lie groups are closely related to affine connections, as they are used to represent the symmetries and transformations of a space that is equipped with an affine connection. The structure of a Lie group allows for the study of the properties and behavior of an affine connection.
An affine connection is a mathematical concept that describes how tangent spaces are connected on a smooth manifold. It is used to define parallel transport, which is a way of moving vectors along curves on a manifold while maintaining their direction.
Lie groups and affine connections are used in physics to study the symmetries and transformations present in physical systems. They are particularly useful in the study of general relativity, where they are used to describe the structure of spacetime.
Yes, affine connections can be defined on any type of space that is equipped with a smooth structure. This includes both finite-dimensional spaces, such as Euclidean spaces, and infinite-dimensional spaces, such as function spaces.