  Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) <
\dim(N)$, and let $M$ compact. Then shape space in this work is either the
manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold
of unparametrized immersions of $M$ in $N$. We investigate the Sobolev
Riemannian metrics on shape space: These are induced by metrics of the
following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \g(P^f h,
k)\, \vol(f^*\g)$$ where $\g$ is some fixed metric on $N$, $f^*\g$ is the
induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to
the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint,
bijective scalar pseudo differential operator of order $2p$ depending smoothly
on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where
$\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\bar g$. For
these metrics we compute the geodesic equations both on the space of immersions
and on shape space, and also the conserved momenta arising from the obvious
symmetries. We also show that the geodesic equation is well-posed on spaces of
immersions, and also on diffeomorphism groups. We give examples of numerical
solutions.
