# Smooth approximation to a continuous curve

Let $\gamma: [0,1] \rightarrow M$ be a continuous curve in a smooth manifold $M$. Is there a standard way to approximate $\gamma$ by a smooth curve? My thought was to look at every point $p$ where $\gamma$ is not smooth, consider a coordinate chart $(U, \phi)$ containing $p$ and smoothen $\gamma \cap U$. Can this be made precise?

• There are recent questions asking about smoothing a piecewise linear curve; or piecewise smooth; for these you can make modifications at the finite set of points in which the curve is not smooth. But in general, there's no reason you should be able to make "small, local modifications": what if the curve is big and gross, like a space-filling curve? Your best bet then is probably to do a convolution with a (very small) smooth bump function. This will smooth your curve to a curve $\eta$, and will not be "far" in the sense that you can make $\text{max }d(\gamma(t),\eta(t))$ as small as you want – user98602 Sep 2 '15 at 3:45
• Do you have an example or reference on that? – user265669 Sep 2 '15 at 3:51
• I suggest you look at, say, Bredon's Topology and Geometry, theorem II.11.8: Suppose $M^m$ and $N^n$ are smooth manifolds with $N^n$ compact metric (by this we just mean we have fixed a metric i.e. a distance function, not a Riemannian metric). Let $A \subset M^m$ be closed. Let $f : M^m \to N^n$ be a map with $f|_{A}$ smooth. Then for any given $\epsilon > 0$, there exists a map $h : M^m \to N^n$ such that 1) $h$ is smooth 2) $d(h(x), f(x)) < \epsilon$ for all $x \in M^m$ 3) $h|_{A} = f|_{A}$ 4) $h \simeq f$ by an $\epsilon$-small homotopy (rel $A$) – Pedro Sep 3 '15 at 0:52
• So basically you can homotope your continuous curve to a smooth one, and you can do so without changing it on a subset where it's already smooth. So, for example, trivially $\gamma$ is smooth on $\{0, 1\}$, hence you may homotope it to a smooth curve without changing the endpoints. – Pedro Sep 3 '15 at 0:55