When I look up the definition of a Lie group I find that it is a differential manifold. When I look up the definition of a manifold I find that it is a space that is locally Euclidean. My understanding is that a manifold is not required to have a metric tensor or distance measure so "Euclidean" cannot be referring to the Pythagorean theorem for triangles. So what does Euclidean mean? I look up the definition of a Euclidean space and find that it is a space defined through axiomatic theory. So I put all of these statements together to obtain the definition of a Lie group? We invent an axiomatic theory to obtain a manifold, then arrange for it to be differentiable (whatever that means) and now we have a Lie group. This makes no sense to me. Can somebody please give more understandable definitions of Lie group, manifold, and Euclidean space?
I took a first course in abstract algebra where a group was defined without any mention of a manifold. It seems to me that reference to a manifold in the definition of a Lie group is unnecessary and makes the definition unnecessarily difficult to understand. Even if so, I am still looking for an easy-to-understand definition of a manifold.
A manifold is a space that, locally, that is in the neighborhood of any point, is isomorphic to Rn,m in general, depending on the signature. So, if m or n is 0, it's Euclidian, else it's Lorentzian.
A manifold most certainly has a metric, precisely for the reason mentioned: Euclidian does refer to the generalization of the Pythagorean theorem. Of course , if the manfold is Lorentzian, then the distance between two points uses the Lorentzian metric.
A Lie group is, first of all, a group. The ``Lie'' qualifier means that the set of elements of a Lie group is a manifold, so the group operations correspond to operations consistent with the geometry of the manifold.
So it just doesn't make sense to speak of a Lie group that isn't a manifold.
A useful introduction to Lie groups is here: https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/pages/lecture-notes/
When people speak about Euclidean spaces they could mean quite different things depending on the context. One could mean classical geometry (with Pythagorean theorem and so forth) that can be axiomatized. This is definitely not what you need to understand manifolds.
A bit closer description is that a Euclidean space is something complete and having constant zero curvature. This puts it apart from constant positive curvature complete spaces (= spheres), and hyperbolic spaces having constant negative curvature. However, if we forget the metric structure, hyperbolic and Euclidean spaces of the same dimension are indistinguishable, but they are different from spheres. If you remove a point from a sphere, you can identify it with a Euclidean space via a stereographic projection. Also, spheres are compact and Eucledian spaces of positive dimension are not.
The identification via a stereographic projection is an example of a homeomorphism. Together with the notion of compactness, mentioned above, it belongs to the realm of topology, a very elegant and beautiful field of mathematics, where manifolds originate. Also, the word "locally", at least to me, makes sense only when you speak of topological spaces. Their definition is not a hard one, but takes some practice to become comfortable with.
For example, a one-dimensional Euclidean space is a real line with the topology where open sets are unions of open intervals. An n-dimensional Euclidean space is just a cartesian product of n copies of one-dimensional ones. A topological manifold X of dimension n is a topological space that is locally homeomorhpic to a Euclidean space, which means that for any point of p of X, there is a neighborhood U of p (that is an open set in X containing x) homeomorphic to an n-dimensional Euclidean space. What this implies, is that around every point the space X can be described in terms of coordinates in the neighbourhood, which is referred as a chart. Spheres are examples of manifolds, in particular, in dimension=two atlases of maps (say of hemispheres) give charts of the surface of the Earth.
A differentiable manifold is a topological manifold where charts can be chosen in such a way that transitions from one to another are smooth in the respective local coordinates (usually, but not always, this means infinitely differentiable). It is an extremely interesting phenomenon, that there are topological manifolds, which has no differentiable structure. Moreover, there can be more than one non-equivalent differentiable structure on the same topological manifold. For example, it is only in dimension four, a Euclidean space can have exotic smooth structures. However, it is not known if any of those survive after a one-point compactification, i.e. if there are exotic smooth structures on the four-dimensional sphere -- this is the remaining part of the Poincaré conjecture.
A Lie group is a type of topological group, where the underlying topological space is a differentiable manifold, and the binary operation and the inversion operation are both smooth. There are many topological groups, say in number theory or algebraic geometry that are not Lie groups, however, any abstract group can be given a structure of a discrete topology, where each point is its one neighborhood, i.e. it can be said that it is locally a zero-dimensional Eucledian space...
Anyhow, I think that familiarizing yourself with concrete examples would be much more useful than memorizing all the general definitions. For example, the Lie group U(1) is diffeomorphic to a circle, and it can be very conveniently imagined as a space of unit absolute value complex numbers, the operation is just the multiplication of such numbers. Similarly, SU(2) is a three-dimensional sphere as a manifold and can be seen as an absolute value one quaternions.
In which sense Lie groups are better than abstract groups? Well, they and their actions on spaces can be linearized. The corresponding concept is that of a Lie algebra.
In 1952, Andrew M. Gleason proved that ALL locally Euclidean topological groups were Lie groups taking a big step into Hilbert's fifth problem. Then a bunch of wonderful things happened that for me ends starting with [1] Robert E, Gompf, “An Infinite Set of Exotic R4’s”, J. Differential Geom. 21(2): 283-300 (1985). DOI: 10.4310/jdg/1214439566,
https://projecteuclid.org/journals/journal-of-differential-geometry/volume-21/issue-2/An-infinite-set-of-exotic-mathbfR4s/10.4310/jdg/1214439566.full
Thank you all for the inputs. What I learned from this thread is that I have a lot to learn. That Lie groups are necessarily intimately connected with manifolds (cannot be defined without referring to manifolds) is something that I was not expecting. I also learned that there will not be any one-paragraph explanation that is simple enough for me to understand it. I need to get more educated. Thank you for the guidance towards that education.
Euclidean space means (real) vector space of finite dimension (we speak of Hilbert spaces, Banach spaces...in the case of infinite dimension).
To define a manifold, we need topology (space locally homeomorphic to IR^n). To define a Lie group we need topology (manifold structure) and algebra (group structure) with compatibility between the two structures.
An important theorem shows that any closed subgroup of the linear group (group of invertible matrices) is a Lie group. A nice introduction to these subject is the book by Loring Tu, An Introduction to Manifolds.
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