# For manifolds of the same dimension, are submersions equivalent to immersions?

My book is An Introduction to Manifolds by Loring W. Tu. Immersions and submersions are defined here.

1. Let $$A$$ and $$B$$ be manifolds with the same dimension $$d$$, and let $$G: A \to B$$ be a smooth map. I think that for each $$p \in A$$, $$G$$ is a submersion at $$p$$ if and only if $$G$$ is an immersion at $$p$$ because $$G_{*,p}$$ is a homomorphism of vector spaces of the same finite dimension $$d$$.

Is this correct? If so, then I have 2 follow-up questions.

1. Can we restate Remark 8.12 of the inverse function theorem as follows?

$$F$$ is a local diffeomorphism at $$p$$ if and only if any of two equivalent conditions hold:

• $$F$$ is a submersion at $$p$$,

• $$F$$ is an immersion at $$p$$.

2. In this question What does it take for a smooth homeomorphism to be a diffeomorphism?, can we say submersion instead of immersion given that homeomorphism of smooth manifolds implies same dimension, as with diffeomorphism?

• In some ways, I think one would expect immersion since what it takes for a smooth topological embedding to be a smooth embedding, as defined here, is being an immersion.

• I was actually surprised to see immersion instead of submersion. Since submersions are open maps, I initially thought of submersion as the smooth analogue for "open map", in the sense that just as we have, for a bijective continuous map $$g$$ of topological spaces, that $$g^{-1}$$ is continuous if and only if $$g$$ is open, I thought that we would have, for the $$f$$ in the question, $$f^{-1}$$ is smooth if and only if $$f$$ is a submersion.

## 1 Answer

You are correct on all three points.

The differential is a map between tangent spaces. If both tangent spaces have the same (finite) dimension, then an injective map is also a surjective map and is thus an isomorphism.

A local diffeomorphism between manifolds of the same dimension is indeed just an immersion or a submersion, as injectivity, surjectivity, and being an isomorphism on the level of tangent spaces are all equivalent.

If we have a smooth homeomorphism, your linked answer shows that it is a diffeomorphism if and only if it is an immersion. We know that a homeomorphism must be a map between manifolds of the same dimension, so here immersion is equivalent to submersion.

• Oh right so now that I know that homeomorphisms like diffeomorphisms imply the same dimension, we can simply replace immersion for submersion? Thanks! – user636532 Jul 20 '19 at 12:16
• @SeleneAuckland Yup, given a smooth map between equal dimension manifolds, any time you see “immersion” feel free to say “submersion” or “local diffeomorphism” instead. – Santana Afton Jul 20 '19 at 12:22