I'm reading Peter Petersen's Riemannian Geometry of soul theorem.
I know if a Riemannian manifold with positive sectional curvature, its soul is a point.
However I don't know why soul is a point implies $M$ is diffeomorphic to $\mathbb{R}^n$?
I guess:
(1) A Riemannian manifold whose totally geodesic submanifold is a point is diffeomorphic to $\mathbb{R}^n$?
(2) A contractible differential manifold is diffeomorphic to $\mathbb{R}^n$?
Thanks.
Best Answer
Both your assumptions 1. and 2. are wrong. A point is always trivially a totally geodesic submanifold. And a contractible manifold need not even be homeomorphic to Euclidean space.
This is really just a matter of unwinding the definitions. Recall the definition of a soul of an $n$-dimensional Riemannian manifold $(M^n, g)$: it's a closed totally convex, totally geodesic embedded submanifold whose normal bundle is diffeomorphic to $M$. What is the normal bundle $T^{\perp} S$ of a submanifold $S \subset (M, g)$? Well, to know that we have to know what each of its fibers are (actually, a bundle obviously carries more structure than that - it has a natural smooth atlas and topology, but for the purposes of your question you can just take this as granted). And what are its fibers at each point $p$? They are the vector spaces $T^{\perp}_p S$ defined by:
$$T^{\perp}_p S = \{v \in T_p M \ \vert \ g(v, s) = 0 \ \text{for all } s \in T_p S \}$$
The total space of the normal bundle is then just:
$$T^{\perp} S = \bigcup_{p \in S} T^{\perp}_p S$$
So what happens when $S = \{p \}$ is just a single point? Then obviously its tangent space at an arbitrary $q \in S$ (of course, there is only one such $q$, which is $p$, but bear with me here) is just the zero dimensional vector space given by $$T_q S = \{0_{T_p M}\}$$ So it's clear that: $$T^{\perp}_q S = T_p M$$ Hence, the total space of the normal bundle of $S$ is given by:
$$T^{\perp} S = \bigcup_{q \in S} T_q S = T_p M$$
Now, $T_p M$ is an $n$-dimensional vector space and obviously diffeomorphic to $\mathbb{R}^n$. So if a soul of $M$ is a single point, then by definition $M$ must be diffeomorphic to the normal bundle of that single point, which as we just saw is diffeomorphic to $\mathbb{R}^n$. Thus $M$ is diffeomorphic to $\mathbb{R}^n$.
If you still don't understand it after this, I'd suggest thinking about some concrete examples. Can you explicitly find a soul or find the normal bundle for commonly known regular surfaces of $\mathbb{R}^3$? Can you find an example where the soul of $M$ cannot ever be a point? That kind of thing, start small.