  We present the $d+1$ formulation of Einstein-scalar-Gauss-Bonnet (ESGB)
theories in dimension $D=d+1$ and for arbitrary (spacelike or timelike)
slicings. We first build an action which generalizes those of
Gibbons-Hawking-York and Myers to ESGB theories, showing that they can be
described by a Dirichlet variational principle. We then generalize the
Arnowitt-Deser-Misner (ADM) Lagrangian and Hamiltonian to ESGB theories, as
well as the resulting $d+1$ decomposition of the equations of motion. Unlike
general relativity, the canonical momenta of ESGB theories are nonlinear in the
extrinsic curvature. This has two main implications: (i) the ADM Hamiltonian is
generically multivalued, and the associated Hamiltonian evolution is not
predictable; (ii) the "$d+1$" equations of motion are quasilinear, and they may
break down in strongly curved, highly dynamical regimes. Our results should be
useful to guide future developments of numerical relativity for ESGB gravity in
the nonperturbative regime.

General Relativity and Quantum Cosmology