Correction: Bates et al. Vector Fields and Differential Forms on the Orbit Space of a Proper Action. Axioms 2021, 10, 118

There was an error in the original article [1]. In the proof of lemma 5, Equation (19) is incorrect. A correction has been made to the proof of Lemma 5 in Section 2 Vector fields on $M/G$ to the fifth, sixth and seventh paragraphs.

The corrected fifth paragraph is:

The module $\mathfrak{X}{\left(B\right)}^H$ of H invariant smooth vector fields on B is finitely generated by polynomial vector fields, see [16], and we denote a generating set by ${\left\{X_j\right\}}_{j=1}^N$. Hence, every H invariant smooth vector field $X_B$ on B is of the form $X_B={\sum }_{j=1}^N{h}_j{X}_j$ for some $h_1,\dots ,h_n\in C^{\infty }\left(B\right)$. Similarly, every K invariant smooth vector field on $B_{\left(K\right)}$ can be written as ${\sum }_{j=1}^N{f}_j{X}_j$, where $f_j\in C^{\infty }\left(B_{\left(K\right)}\right)$. Since $B_{\left(K\right)}$ is open and dense in B, a generic $f\in C^{\infty }\left(B_{\left(K\right)}\right)$ need not extend to a smooth function on B. Therefore a generic vector field on $B_{\left(K\right)}$ need not extend to a smooth vector field on B. On the other hand, the H invariant vector field $X_{B_{\left(K\right)}}$ on $B_{\left(K\right)},$ is obtained above from a smooth bounded vector field Y on $B/H$. Therefore

$$X_{B_{\left(K\right)}}\left(\mathbf{x}\right)=\sum _{j=1}^N{k}_{j\mid B_{\left(K\right)}}\left(\mathbf{x}\right)X_j\left(\mathbf{x}\right),$$

(19)

for each $\mathbf{x}\in B_{\left(K\right)}\subseteq B\subseteq {\mathbb{R}}^n,$ where every $k_j\in C^{\infty }\left({\mathbb{R}}^n\right)$, and $k_{j\mid B_{\left(K\right)}}$ is the restriction of $k_j$ to $B_{\left(K\right)}$.

Since $B_{\left(K\right)}$ is open and dense in B, we may define

$$X\left(\mathbf{x}\right)=\left\{\begin{array}{cc}{X}_{B_{\left(K\right)}}\left(\mathbf{x}\right)& \mathrm{if}\mathbf{x}\in B_{\left(K\right)}\hfill \\ {\sum }_{i=1}^N{lim}_{k\to \infty }{k}_j\left({\mathbf{x}}_k\right)X_j\left({\mathbf{x}}_k\right),& \mathrm{where}\left\{{\mathbf{x}}_k\right\}\subseteq B_{\left(K\right)}\mathrm{and}\mathbf{x}=\underset{k\to \infty }{lim}{\mathbf{x}}_k\in B\backslash B_{\left(K\right)},\hfill \end{array}\right.$$

(20)

provided that ${lim}_{k\to \infty }{k}_j\left({\mathbf{x}}_k\right)$ exists and is unique. Since the vector fields $X_j\left({\mathbf{x}}_k\right)$ are smooth on B,

$$\underset{k\to \infty }{lim}{k}_j\left({\mathbf{x}}_k\right)X_j\left({\mathbf{x}}_k\right)=\left(\underset{k\to \infty }{lim}{k}_j\left({\mathbf{x}}_k\right)\right)X_j\left(\underset{k\to \infty }{lim}{\mathbf{x}}_k\right)\mathrm{for}\mathrm{every}j=1,\dots ,N.$$

The corrected sixth paragraph is:

Moreover, since $B_{\left(K\right)}$ is open and dense in $B,$ it is open and dense in $\overline{B}$, the closure of B. In Equation (19), each function $k_{j\mid B_{\left(K\right)}}$ is the restriction to $B_{\left(K\right)}\subseteq B\subseteq {\mathbb{R}}^n$ of a smooth function $k_j$ on ${\mathbb{R}}^n$. Moreover, the choice of polynomial basis ${\left\{X_j\right\}}_{j=1}^N$ ensures that the right-hand side of Equation (19) extends to the closure $\overline{B}$ of B. Hence all the the limits ${lim}_{k\to \infty }{k}_j\left({\mathbf{x}}_k\right)$ in Equation (20) exist, and $X\left(\mathbf{x}\right)$ is defined for all $\mathbf{x}\in B$.

The corrected seventh paragraph is:

We need to show that this definition of X on B depends only on $X_{B_{\left(K\right)}}$. Since each $k_j$ is continuous on B and its first partial derivatives are bounded on $\overline{B}$, it follows that $k_j$ are uniformly continuous on $\overline{B}$. In particular, if $c:[0,1]\to \overline{B}$ is a smooth curve, such that $c\left([0,1)\right)\subseteq B_{\left(K\right)}$ and $c\left(1\right)\in B$, then

$$k_{j\mid B}\left(c\left(1\right)\right)=k_{j\mid B}\left(c\left(0\right)\right)+{\int }_0^1\frac{\partial k_{j\mid B}}{\partial t}\left(c\left(t\right)\right)dt=k_{j\mid B_{\left(K\right)}}\left(c\left(0\right)\right)+{\int }_0^1\frac{\partial k_{j\mid B_{\left(K\right)}}}{\partial t}\left(c\left(t\right)\right)dt.$$

Thus, the values of $k_j$ on B are uniquely determined by $k_{j\mid B_{\left(K\right)}}$. Repeating this argument for all the first-order partial derivatives of $k_j$, we deduce that the first-order partial derivatives of $k_j$ on B are uniquely determined by $k_{j\mid B_{\left(K\right)}}$ and its first partial derivatives. Continuing this process for every partial derivative of every order shows that the restriction of $k_j$ to B is uniquely determined by $k_{j\mid B_{\left(K\right)}}$.

The authors apologize for any inconvenience caused and state that the scientific conclusions are unaffected. The original article has been updated.