For an open subset $\Omega\subset \mathbb R^n$ one can define the Sobolev space
$$H^1(\Omega):=W^{1,2}(\Omega)=\{u \in L^2(\Omega) \, \vert \, \partial u \in L^2(\Omega)\}.$$
Is there a "simple" way to introduce the Sobolev space $H^1$ on a compact, Riemannian manifold? Is it equivalent tot he Euclidean case?
Sure. If $M$ is a compact manifold, and $k$ is a non-negative integer, then $H^k(M)$ is the space of function $u\in L^2(M)$ with the property that for any $\ell$ smooth vector fields $X_1,\cdots, X_\ell$ on $M$, with $\ell\leq k$, we have $X_1\cdots X_\ell u\in L^2(M)$. For a reference, you can see Michael Taylor's PDE I text.