  Using a covariant geometric approach we obtain the effective bending
couplings for a 2-dimensional rigid membrane embedded into a
$(2+D)$-dimensional Euclidean space. The Hamiltonian for the membrane has three
terms: The first one is quadratic in its mean extrinsic curvature. The second
one is proportional to its Gaussian curvature, and the last one is proportional
to its area. The results we obtain are in agreement with those finding that
thermal fluctuations soften the 2-dimensional membrane embedded into a
3-dimensional Euclidean space.
