I was trying to prove that the horn torus and the spindle torus are not manifolds by definition(locally diffeomorphic to some Euclidean space.). I have no idea how to do this, but I attempted it in the following way:
I failed to show that a "slice" of manifold is a manifold itself. By a slice I mean, if the manifold $X\in \mathbf R^n$, then we set the coordinates $x_i=0$ for some $i$ where $0\leq i\leq n$. I feel this would work but I have no idea how to prove it.
Then I look at the cross section for the tori I mentioned(this is equivalent to taking a slice.). For the horn torus, you have two circles touching each other. And the fore spindle torus, you have two circles intersecting each other. Since a circle is locally diffeomorphic to $\mathbf R^1$, the two circles better have to be diffeomorphic to $R^1$. Otherwise we get the result we want.
I tried to proof that a neighbourhood around the point they touch(or intersect) cannot be locally diffeomorphic to $\mathbf R^1$. I tried to do it by contradiction. However I'm stuck on finding a contradiction...
Any thoughts?
P.S. Under the request of Sam Lisi, here are the definitions or horn and spindle tori:
A torus is the set of points in $\mathbf R^3$ at a distance $b$ from the circle of radius a in the $xy$ plane. This is like you put a circle with radius $b$ in the $yz$ plane centred at $(a,0,0)$. Then you make it orbit around the origin and you get a torus.
A horn torus is when $a=b$. If you take a cross-section, you'll find that it's two circles touching each other at one point. When $a<b$, it's called a spindle torus. The cross-section would look like two circles intersecting with each other at two pionts.
There are pictures in this wikipedia article that might help you visualise the horn and spindle tori: http://en.wikipedia.org/wiki/Torus
Often the best way to show a $k$-dimensional subset $M$ of $\mathbb R^n$ is not a submanifold is this: Show that there's some point $p\in M$ so that no neighborhood is a graph of a smooth function over any of the standard coordinate $k$-planes.
In the case of your horn torus, at that central pinch point, the vertical line test will fail for any choice of the 3 standard coordinate ($2$-)planes ($xy$, $yz$, $xz$). Indeed, by symmetry, it fails for every plane, but you don't need to show this. (Intuitively, if the surface were to have a tangent plane at this point, by rotational symmetry, it would have infinitely many.)