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I'm doing some reading on isometric actions and the book I'm following just proved that one has diffeomorphisms

$\mathbb{S}^{n} \cong \mathrm{SO}(n+1)/ \mathrm{SO}(n)$,

$\mathbb{R}P^{n}\cong \mathrm{SO}(n+1)/S(O(n)\times O(1))$.

Then he proves analogous statements for the grassmanian manifolds but then he completely changes the subjects and leaves these just as examples. I was wondering if there are any immediate applications or conclusions one can draw from having these descriptions of the spheres, the projective spaces and the grassmanians. For instance, the book mentions that this allows us to see them as homogeneous spaces. What else can (easily) be said?

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Number theorists are quite big on these type of constructions. When you have a group action on a space, algebra comes into play, and you can look at discrete subgroups that might have some nice arithmetic properties. See for more background, and check out the Wikipedia entries for spherical functions and Gelfand pairs (if you dare!). – John M Aug 15 '11 at 12:22
Using $\mathbb{S}^{n} \cong \mathrm{SO}(n+1)/ \mathrm{SO}(n)$ gives a way of showing that $\mathrm{SO}(n)$ is connected for every $n$ (by induction). – Sam Aug 15 '11 at 12:24
Here's a similar game you can play: The upper-half of the complex plane $\mathbb H \cong SL(2,\mathbb R)/SO(2)$. This can help you answer the question: What is the topology of $SL(2,\mathbb R)$? The quotient map $SL(2,\mathbb R) \rightarrow SL(2,\mathbb R)/SO(2) \cong \mathbb H$ is a fiber bundle, which must be trivial since $\mathbb H$ is contractible. Thus, topologically speaking, $SL(2,\mathbb R)$ is homeomorphic to $SO(2) \times \mathbb H$, which in turn is homeomorphic to the open torus. – John M Aug 15 '11 at 12:35
Here's another example: the Grassmanian has the diffeomorphism $Gr(k,n) \cong O(n)/(O(k) \times O(n-k))$. Since the dimension of $O(n)$ is $n(n-1)/2$, we get the dimension of the Grassmanian is $k(n-k)$. – John M Aug 15 '11 at 13:19
All of these comments are excellent and the only reason I'm accepting the answer below is because, well it's in the form of an answer. Thank you very much. – Sak Jan 29 '13 at 1:40
up vote 6 down vote accepted

You gain several things.

  1. A method for computing their topology (almost). Given an inclusion of (say) compact Lie groups $H\subseteq G$, there is a spectral sequence which computes the cohomology groups (and often ring structure) as well as characteristic classes (for the tangent bundle of G/H). Often in practice (but not always, hence the above "almost") the differentials are actually computable.

For spheres, the cohomology ring and characteristic classes are easy to compute. For projective spaces, this information is harder to compute.

  1. (Again, I'm assuming we're working with compact Lie groups) A metric of nonnegative sectional curvature. Until recently (as in, the last 5 years or so), they only known and easy to apply method for constructing metrics of nonnegative sectional cuvature was by starting with $G$ with biinvariant metric and quotienting out by some subgroup.

  2. Writing as a homogeneous space helps you find more isometric actions. As an example, consider $S^7$. This can be expressed as a homogeneous space in 4 different ways: $S^7 = SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/G_2$.

The $SU(3)$ in $SU(4)$ is not maximal - one has $SU(3)\subseteq U(3)\subseteq SU(4)$. It turns out that the extra circle factor in the $U(3)$ descends to a well defined isometric action on $S^7$ - we've just discovered the Hopf fibration $S^1\rightarrow S^7\rightarrow \mathbb{C}P^3$.

Likewise, the $Sp(1)$ in $Sp(2)$ is not maximal - one has $Sp(1)\subseteq Sp(1)\times Sp(1)\subseteq Sp(2)$. It turns out the extra $Sp(1) = S^3$ factor gives a well defined isometric action on $S^7$. We've just discovered the Hopf fibration $S^3\rightarrow S^7\rightarrow S^4$.

(The funny $Spin(7)/G_2$ doesn't give rise to any new "exotic" actions on $S^7$, but it does help one to understand the spin representation of $Spin(7)$ and also what $G_2$ actually is).

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