  A compact Riemannian manifold is associated with geometric data given by the
eigenvalues of various Laplacian operators on the manifold and the triple
overlap integrals of the corresponding eigenmodes. This geometric data must
satisfy certain consistency conditions that follow from associativity and the
completeness of eigenmodes. We show that it is possible to obtain nontrivial
bounds on the geometric data of closed Einstein manifolds by using semidefinite
programming to study these consistency conditions, in analogy to the conformal
bootstrap bounds on conformal field theories. These bootstrap bounds translate
to constraints on the tree-level masses and cubic couplings of Kaluza-Klein
modes in theories with compact extra dimensions. We show that in some cases the
bounds are saturated by known manifolds.
