  We prove that some Riemannian manifolds with boundary under an explicit
integral pinching are spherical space forms. Precisely, we show that
3-dimensional Riemannian manifolds with totally geodesic boundary, positive
scalar curvature and an explicit integral pinching between the $L^2$-norm of
their scalar curvature and the $L^2$-norm of their Ricci tensor are spherical
space forms with totally geodesic boundary. Moreover, we prove also that
4-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe
invariant and an explicit integral pinching between the total integral of their
$(Q,T)$-curvature and the $L^2$-norm of their Weyl curvature are spherical
space forms with totally geodesic boundary. As a consequence of our work, we
show that a certain conformally invariant operator which plays an important
role in Conformal Geometry has a trivial kernel and is non-negative if the
Yamabe invariant is positive and verifies a pinching condition together with
the total integral of the $(Q,T)$-curvature. As an application of the latter
spectral analysis, we show the existence of conformal metrics with constant
$Q$-curvature, constant $T$-curvature, and zero mean curvature under the latter
assumptions.
