Consider a dynamical system $\Sigma$ on a manifold $M$ of dimension $d$ embedded in $\mathbb{R}^n$, where $d<n$. Let $x\in M$ be an equilibrium point and suppose we wish to determine the stability of $x$. Let $L$ denote the linearization matrix of $\Sigma$ at $x$. Suppose that there are eigenpairs $(\lambda_i,v_i)$ of $L$ with $\lambda_i<0$ such that $\{v_1,\ldots,v_n\}$ is a basis for the tangent space $T_xM$ at $x$. Is there a version of the Hartman-Grobman theorem which allows us to conclude that $x$ is asymptotically stable?
In other words, if the linearization is done in the embedding space, can we (for the purpose of stability analysis) consider the linearized system at $x$ to be constrained to the tangent space $T_xM$?