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The map $\varphi : M \to N$ is an embedding if $\varphi$ is a homeomorphism, an immersion, and $\varphi(M) \subset N$, where $M$ and $N$ are manifolds.

The curve $\alpha(t)=(t^3-4t,t^2-4)$ has a self-intersection for $t=2$ and $t=-2$ (because $\alpha(2)=\alpha(-2)=(0,0)$).

Why is this curve not an embedding? I do not know how $\alpha(\mathbb R) \subset \mathbb R^2$ does not hold. Because the curve has a self-intersection, there's some $t$ such that $\alpha(t) \not\in \mathbb R^2$?

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    I think your quoted definition is the definition of _immersion_. In an embedding, $M$ also has to be homeomorphic to $\varphi(M)$, or $\varphi$ has to be injective, or any of a number of other different, but equivalent, criteria. – 2017-01-02
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    @Newdayrising: Your stated definition of "embedding" is far from standard. For example, a space-filling curve or a constant mapping is an "embedding" in this sense. – 2017-01-02
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    @Arthur I did not copy do Carmo's textbook definition propertly, but I understand it now and it answered my question: since $\varphi$ must also be a homeomorphism from $M$ to $\varphi(M)$ it must be bijective (injective and surjective), in addition to being continuous and having continuous inverse. – 2017-01-02
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    @Newdayrising: For what it's worth, the stated definition is still non-standard. In the setting of differential geometry, one normally assumes $\varphi$ is a smooth immersion (the tangent mapping has maximal rank at each point) and a homeomorphism (hence a diffeomorphism) onto its image. (You're correct that the problem with your $\varphi$ is it's not bijective. Note, however, that an injective immersion can fail to be an embedding.) – 2017-01-02

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In general you want the map to be at least injective. Also the image to be smooth could be a good thing to ask. Even tho $$\alpha'(t)=(3t^2-4,2t) \not= (0,0)$$ unfortunatley the map is not injective because $\alpha(2)=\alpha(-2)$.

Loosely speaking what you really want is something like a clone of $\mathbb R$ in $\mathbb R^2$, then of course you don't want self intersection and you want it to be smooth. A general way to ask for this is to ask for injectivity and for the injectivity of the induced maps (one per point) between the tangent spaces.