  The critical behavior of semi-infinite $d$-dimensional systems with
$n$-component order parameter $\bm{\phi}$ and short-range interactions is
investigated at an $m$-axial bulk Lifshitz point whose wave-vector instability
is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. The associated
$m$ modulation axes are presumed to be parallel to the surface, where $0\le
m\le d-1$. An appropriate semi-infinite $|\bm{\phi}|^4$ model representing the
corresponding universality classes of surface critical behavior is introduced.
It is shown that the usual O(n) symmetric boundary term $\propto \bm{\phi}^2$
of the Hamiltonian must be supplemented by one of the form $\mathring{\lambda}
\sum_{\alpha=1}^m(\partial\bm{\phi}/\partial x_\alpha)^2$ involving a
dimensionless (renormalized) coupling constant $\lambda$. The implied boundary
conditions are given, and the general form of the field-theoretic
renormalization of the model below the upper critical dimension
$d^*(m)=4+{m}/{2}$ is clarified. Fixed points describing the ordinary, special,
and extraordinary transitions are identified and shown to be located at a
nontrivial value $\lambda^*$ if $\epsilon\equiv d^*(m)-d>0$. The surface
critical exponents of the ordinary transition are determined to second order in
$\epsilon$. Extrapolations of these $\epsilon$ expansions yield values of these
exponents for $d=3$ in good agreement with recent Monte Carlo results for the
case of a uniaxial ($m=1$) Lifshitz point. The scaling dimension of the surface
energy density is shown to be given exactly by $d+m (\theta-1)$, where
$\theta=\nu_{l4}/\nu_{l2}$ is the anisotropy exponent.
