Generalized level crossings and tangencies of a random field with smooth sample functions

Tangencies and level crossings of a random field X:m+x[Omega]-->n (which is not necessarily Gaussian) are studied under the assumption that almost every sample path is continuously differentiable. If n=m and if the random field has uniformly bounded sample derivatives and uniformly bounded densities for the distributions of the Xl, then for a compact K[subset of]m+ and any fixed level, the restriction to K of almost every sample path has no tangencies to the level and at most finitely many crossings. The case of n[not equal to]m is also examined. Some generic properties, which hold for a residual set of random fields, are analyzed. Proofs involve the concepts of regularity and transversality from differential topology.