Intuitively what are submanifolds?

Question

I'm a physics student and my understanding on differential geometry and manifolds is based on general relativity. Can you explain the concept of (smooth) submanifold intuitively? Is a smooth manifold embedded in higher dimensional Euclidean space a submanifold? If that manifold is curved then its geodesics are not geodesics of the Euclidean space but I think it would make sense to define that geodesics of a submanifold also have to be geodesics of the manifold. Does some definition include that?

Answer 1

If $M$ is a smooth manifold, then a submanifold of $M$ is a subset $N \subset M$ that is itself a manifold with the smooth structure it inherits from $M$. By the Nash embedding theorem, any manifold is a submanifold of Euclidean space, so in some sense that's the only case we need to consider.

Everyone's favorite submanifold is the sphere $S^n = \{ x \in \mathbb{R}^{n+1} \mid |x| = 1\} \subset \mathbb{R}^{n+1}$. For other cheap examples, you can look at the zeros of polynomials on $\mathbb{R}^n$, which will "usually" define a smooth submanifold.

The sphere shows you that the geodesics of the submanifold (great circles) do not need to be geodesics of the ambient manifold (straight lines). This can happen, however, if the inclusion map $j : N \to M$ is a Riemannian embedding. That is a much stronger condition than for $N$ to be a submanifold of $M$. The second fundamental form measures the failure of this condition; if it is zero, then the geodesics of the submanifold are geodesics of the ambient manifold, otherwise they will differ in general.