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The image $f(M)$ is called a **submanifold** of $N$ . 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From theorem 2.1, we also find $\\ker f_{*} = \\{0\\}$ . If $f$ is an embedding, $M$ is diffeomorphic to $f(M)$ . Examples will clarify these rather technical points. Consider a map $f: S^1 \\to \\mathbb{R}^2$ in figure 5.11(a). It is an immersion since a one-dimensional tangent space of $S^1$ is mapped by $f_{*}$ to a subspace of $T_{f(p)}\\mathbb{R}^2$ . The image $f(S^1)$ is not a submanifold of $\\mathbb{R}^2$ since $f$ is not an injection. The map $g: S^1 \\to \\mathbb{R}^2$ in figure 5.11(b) is an embedding and $g(S^1)$ is a submanifold of $\\mathbb{R}^2$ . Clearly, an embedding is an immersion although the converse is not necessarily true. In the previous section, we occasionally mentioned the embedding of $S^n$ into $\\mathbb{R}^{n + 1}$ . 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