Diffeomorphism of level sets of functions depending on a parameter

Question

Let $H : \mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function with arguments $(x, \alpha)$, where $\alpha$ is a parameter. Fix a constant $c$ and suppose the set $S_{\alpha_0} := \{x: H(x, \alpha_0) = c\}$ is nonempty and consists only of regular values so that it is an $n-1$-dimensional submanifold. Then $S_\alpha := \{x: H(x, \alpha) = c\}$ is a submanifold for $\alpha$ close to $\alpha_0$. What are some simple conditions on $H$ for $S_\alpha$ to be diffeomorphic to $S_{\alpha_0}$ given that $\alpha$ is close to $\alpha_0$?