> [!definition] > > Let $X, Y$ be $C^p$-[[Manifold|manifolds]] ($p \ge 1$), $f: X \to Y$ be a $C^p$-[[Manifold Morphism|morphism]], and $p \in X$. $f$ is an **immersion at** $p$ if there exists [[Atlas|charts]] $(U, \psi) \in X$ and $(V, \varphi) \in Y$ such that the [[Derivative|derivative]] $Df_{U, V}(\hat p)$ is injective and [[Split Subspace|splits]]. > > If $f$ is an immersion at all $p \in X$, then $f$ is an **immersion**. > [!definition] > > Let $X$ be a $C^p$-[[Manifold|manifold]], $Y \subset X$ with any topology, and $\alg$ be an [[Atlas|atlas]] for $Y$. $Y$ is an **immersed submanifold** if the inclusion map $\iota: Y \to X$ is an immersion. > [!definition] > > Let $X$ be a $C^p$-manifold, $Y \subset X$ be an immersed submanifold, then the inclusion map $\iota: Y \to X$ induces a map > $ > d\iota_p = T_p\iota: T_pY \to T_pX > $ > between the [[Tangent Space|tangent spaces]] at each point, which allows identifying $T_pY$ as a subspace of $T_pX$. Since $\iota$ is an immersion, this map is injective and splits.