TITLE: Geometric interpretation of symmetries of a manifold
QUESTION [7 upvotes]: I have been doing some reading about Lie groups and algebras and stumble across the phrase that those theories "measure" the symmetries of the manifolds, like an analog of the groups that measure the symmetries of objects like a triangle, a polygon etc. I am trying to understand what that means.
I think I would understand better the above with 2 examples

So lets take a 2-d manifold-a sphere. Does this mean that we have infinite symmetries? (how do we compute them mathematically meaning if someone ask how many symmetries a sphere has how do we answer that mathematically)
can you give an example of a manifold that has a finite symmetry

How can I think about the symmetries of a manifold? Thinking about an axes' rotation is probably wrong, so how do we think it geometrically?

REPLY [9 votes]: It's essential to understand just what kind of structure you are looking at the symmetries of. If a smooth manifold M is given a Riemannian metric g, then it is a theorem of Myers and Steenrod that its isometry group (the group of self-diffeomorphisms of M that preserve its metric) is a Lie group. Call this Lie group Isom(M,g).
Indeed, all Lie groups whose dimension is positive contain a continuum of points, so they are certainly infinite.
For a circle, the only invariant of a Riemannian metric is its circumference. So its isometry group will be isomorphic to the group O(2), which topologically consists of two circles. (Think of the set of rotations by any angle, and the set of reflections about any line through the center of the circle.)
For a round sphere Sn in Rn+1, its isometry group is called O(n+1) and has dimension equal to n(n+1)/2.
Then certainly the dimension dim(Isom(M,g)) is a "measure" of how symmetrical (M,g) is.
Among surfaces, a round sphere has an isometry group of dimension equal to 3. A flat torus (the quotient of the plane by a lattice, such as R2/Z2, has an isometry group whose dimension is 2. A compact surface of genus greater than or equal to 2, on the other hand — no matter how symmetrical it is — cannot have an isometry group of positive dimension; instead its isometry group is a finite group.