  We prove a sharp logarithmic Sobolev inequality which holds for compact
submanifolds without boundary in Riemannian manifold with nonnegative sectional
curvature of arbitrary dimension and codimension, while the ambient manifold
needs to have a specific Euclid-like property. Like the Michael-Simon Sobolev
inequality, this inequality includes a term involving the mean curvature. This
extends a recent result of S. Brendle with Euclidean setting.
