Standard Galerkin method reduces the problem
Find $u\in V$ such that $a(u,v) = f(v)$ for all $v \in V$, where $V$ is Hilbert space, $a$ is bilinear form and $f\in V^*$.
to a finite dimensional problem by introducing a $n$-dimensional subspace $V_n\subset V$, then we look for an approximation $u_n$ of the solution $u$ such that
Find $u_n \in V_n$ such that $a(u_n,v) = f(v)$ for all $v \in V_n$.
I would like to replace $V_n$ by a $n$-dimensional manifold $\mathcal{M}_n$ in $V$. So the reduced problem would be
Find $u_n\in \mathcal{M}_n$ such that $a(u_n,v)=f(v)$ for all $v\in T_{u_n}\mathcal{M}_n$, where $T_{u_n}\mathcal{M}_n$ is tangent space to the manifold $\mathcal{M}_n$ at the point $u_n$.
What do we know about this problem? What are the conditions on $\mathcal{M}_n$ for existence of $u_n$? Does $u_n$ converge to $u$ as we make $\mathcal{M}_n$ bigger and bigger ($n\rightarrow \infty$)? What is this method called?(I called it Manifold Galerkin method)
Can you please point me to the literature where they discuss this problem?
Although this is probably not exactly what you are looking for, I would like to point you to the Reduced Basis Method which might contain some valuable insight.
It is designed for parametrized PDEs of the form $$ a\big(u(\mu), v ; \mu \big) = f(v;\mu)$$ which allow a structured way to construct a manifold. However, the whole methods starts from a high-dimensional "classical" Galerkin approximation in a vector space. Then, for parameters $\mu \in \mathcal{D}$ the manifold $$\mathcal{M} = \{ u_\mathcal{N}(\mu_i) \}, \quad u_\mathcal{N}(\mu_i) \in V_\mathcal{N} $$ can be constructed which forms the basis of the subsequent analysis. However, this manifold is immediately turned back into a space through the Gram-Schmidt procedure. For this reduced space, a rather comprehensive theory on the properties of approximations based on a parametrized manifold is available.
To sum things up, the reason why you might no find much on the "Manifold Galerking Method" is that you lose the whole theory based on vector spaces (coercivity, continuity, error bounds, ...) but not get nothing in return. In fact, it seems to be more beneficial to construct a vector space based on your manifold (question remains how to choose this if you have not a parametrized PDE) and find links between the different vector spaces.