I'm learning the concepts of immersions at the moment. However I'm a bit confused when they define an immersion as a function $f: X\rightarrow Y$ where $X$ and $Y$ are manifolds with dim$X <$ dim$Y$ such that $df_x: T_x(X)\rightarrow T_y(Y)$ is injective.
I was wondering why don't we let $f$ be injective and say that's the best case we can get for the condition dim$X <$ dim$Y$(since under this condition we can't apply the inverse function theorem)?
Also does injectivity of $df_x$ inply the injectivity of $f$ (it seems that I can't prove it)?
How should we picture immersion as (something like the tangent space of $X$ always "immerses" into the tangent space of $Y$)?
Thanks for everyone's help!
Best Answer
Think of a particle moving around a figure 8 with nowhere zero speed. This parametric curve gives you an immersion $f\colon\mathbb R \to\mathbb R^2$ that is not injective. If you restrict the domain to make it a bijection (which you can do), the image is not a submanifold but is called an immersed submanifold.