Immersion and Submersion

Let $M$ and $N$ be a smooth manifold and $f:M\to N$ is a smooth map.

Immersion

Definition

For each $x\in M$, if the Jacobian

$$d{f}_x:T_{x}M\to T_{f(x)}N$$

is injective.

Example

¹

Submersion

Definition

For each $x\in M$, if the Jacobian

$$d{f}_x:T_{x}M\to T_{f(x)}N$$

is surjective.

Example

1. $f:{\mathbb{R}}^2\setminus 0\to {\mathbb{S}}^1$ as $f(x,y)=(x,y)/\sqrt{x^2+y^2}$.
2. $\pi :T{\mathbb{S}}^2\to {\mathbb{S}}^2$.

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References

Footnotes

1. Milnor, J. W., & Stasheff, J. D. (2022). Characteristic Classes (Rev. ed.), p.38. ↩