The space $SL\_3(\mathbb{R})/SO\_3(\mathbb{R})$ can be though of as some kind of 5-(real)dimensional generalized upper half-space.


Fix any copy of $SO\_2(\mathbb{R})$ sitting inside $SO\_3(\mathbb{R})$, then the coset space $SL\_3(\mathbb{R})/SO\_2(\mathbb{R})$ can be though of as a bundle over $SL\_3(\mathbb{R})/SO\_3(\mathbb{R})$ whose fibers are all $SO\_3(\mathbb{R})/SO\_2(\mathbb{R}) \cong S^2$, the two-sphere.


My question is whether or not this sphere bundle is *trivial*. By which I mean, is it the case that $SL\_3(\mathbb{R})/SO\_2(\mathbb{R})$ is diffeomorphic to $SL\_3(\mathbb{R})/SO\_3(\mathbb{R})\times S^2$.



Is this sphere bundle over SL3/SO3 trivial?
As you remark, the quotient $SL\_n(R)/SO\_n(R)$ is contractible, and so any smooth bundle over it is smoothly trivial. So, in particular your bundle is diffeomorphic to a product.


The contractibility of $SL\_n(R)/SO\_n(R)$ stems from the Gram-Schmidt orthogonalization process; presumably you can use this to exhibit a specific trivialization of your bundle.


