I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics and physics. I visited a course on Lie groups, and an elementary one on Lie algebras. But I don't fully understand how those theories are being applied. I actually don't even understand the importance of Lie groups in differential geometry.
lectures on differential geometry yau schoen pdf 29
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How do people use Lie groups and Lie algebras? What questions do they ask for which Lie groups or algebras will be of any help? And if a geometer reads this, how, if at all, do you use Lie theory? How is the representation theory of Lie algebras useful in differential geometry?
Here is a brief answer: Lie groups provide a way to express the concept of a continuous family of symmetries for geometric objects. Most, if not all, of differential geometry centers around this. By differentiating the Lie group action, you get a Lie algebra action, which is a linearization of the group action. As a linear object, a Lie algebra is often a lot easier to work with than working directly with the corresponding Lie group.
I consider Lie groups and algebras to be near or at the center of the mathematical universe and among the most important and useful mathematical objects I know. As far as I can tell, they play central roles in most other fields of mathematics and not just differential geometry.
ADDED: I have to say that I understand why this question needed to be asked. I don't think we introduce Lie groups and algebras properly to our students. They are missing from most if not all of the basic courses. Except for the orthogonal and possibly the unitary group, they are not mentioned much in differential geometry courses. They are too often introduced to students in a separate Lie group and algebra course, where everything is discussed too abstractly and too isolated from other subjects for my taste.
Let me also try to exapand Deane Yang's answer and explain the importance of Lie groups in differential geometry. Bernhard Riemann solved the equivalence problem (i.e. the question whether a sphere is locally isometric to plane) by developing Riemann geometry and introducing the crucial invariant - the Riemann curvature. Elie Cartan developed a general method for solving such equivalence problems (see Cartan's equivalence method or Method of moving frames on wikipedia). The notion of Lie group is already explicit there as it represents the symmetries of the geometrical structure one is interested in. This approach was later developed into what is now called Cartan geometry.Informally, these geometries are curved versions of Klein geometries. The story can be told like this:
Lie algebras and representation theory also appear, because the tangent space to $G/H$ can be identified with the homogeneous vector bundle associated to the $G$-representaion $\mathfrakg/\mathfrakh$ (this is one of the linearizations people keep talking about). One can regard the curvature tensor as an element of the tensor product of these and decomposition into irreducible subrepresentations then gives generalizations of Weyl and Ricci curvatures from Riemannian geometry. The Dirac operator of mathematical physics can be thought of as a deRham differential composed with a projection and an intertwining map between certain representations. In fact even such fancy gadgets as Lie algebra cohomology play their role (the keyword being "harmonic curvature").
Large subfields of modern differential geometry hardly ever use Lie group theory, e.g. they are never mentioned (as far as I can see) in Schoen-Yau's "Lectures on Differential Geometry", and their role in comparison geometry is quite modest. Major uses of Lie groups in Riemannian geometry are: 2ff7e9595c
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