# Existence of a smooth map given its differential

Let $$M$$ and $$N$$ be smooth manifolds, let $$p \in M$$ and let $$L:T_p M \rightarrow T_{L(p)} N$$ be a linear map.

I wonder if there always exists a smooth map $$F:M \rightarrow N$$ on a neighborhood of $$p$$ such that $$L$$ is the differential map of $$F$$ at $$p$$?

• Your question is worded a bit ambiguously, by suggesting the existence of this particular differential map $dF_p: T_p M \rightarrow T_{F(p)} N$, you inexplicitly suggest that $F$ is already a map (since you used $F(p)$ to parametrise N – Alon Yariv Sep 25 '19 at 9:47
• I modified to question according your observation. – Juan Alvarado Sep 25 '19 at 10:47

I think yes, as long as the discussion is only at a single point $$p$$. I think you mean $$L:T_p M\to T_q N$$ for some points $$p$$ and $$q$$, since $$L(p)$$ doesn't make sense, unless I've misunderstood.
Using charts which send $$p$$ and $$q$$ to 0, your question becomes "can I find a map between euclidean spaces, the derivative of which is the linear map $$L$$?" Since the derivative of a linear map is itself, you can take $$f = L: \mathbb R^m \to \mathbb R^n$$ (where you use the same chart coordinates to identify the tangent spaces with euclidean spaces) to be the representation of your map in these charts.
Then you can multiply by a bump function which is identically 1 on a smaller neighborhood of $$p$$ (so as not to affect the derivative at that point), in order to define a function on the whole of $$M$$.
yes, suppose $$dim M=m, dimN=n$$. Consider charts $$U$$ and $$V$$ of $$p$$ and $$f(p)$$ that you identify to open neighborhoods of $$0$$ in $$\mathbb{R}^m$$ and in $$\mathbb{R}^n$$ with this identification $$p$$ and $$f(p)$$ are identified to $$0$$. Let $$g$$ be a cut off function defined on $$V$$: there exist neighborhoods $$W\subset W_1\subset V$$ of $$0$$ such that the restriction of $$g$$ to $$W$$ is $$1$$ and $$g=0$$ on $$V-W_1$$. Write $$h=gf$$.