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In this paper Bessel potentials on $C^{\infty }$-Riemannian manifolds (open or bordered) are studied. Let $\mathbf{M}$ be an $n$-dimensional manifold, and $\mathbf{N}$ a submanifold of $\mathbf{M}$ of dimension $k$. Sufficient conditions are given for: 1) the restriction to $\mathbf{N}$ of any potential of order $\alpha$ on $\mathbf{M}$ to be a potential of order $\alpha -\frac{n-k}{2}$ on $\mathbf{N}$ ; 2) any potential of order $\alpha -\frac{n-k}{2}$ on $\mathbf{N}$ to be extendable to a potential of order $\alpha$ on $\mathbf{M}$. It is also proved that for a bordered manifold $\mathbf{M}$ the restriction to its interior ${\mathbf{M}}^i$ is an isometric isomorphism between the...