We consider a second-order hyperbolic equation on an openbounded domain-in Rn, with C1-boundary =г=а=гOuг1=φ;, subject to Neumann boundary conditions on the entire boundary. Here,г0 (unobserved/uncontrolled part) and 1 (observed/controlled part) are relativelyopen subsets of ¡. The principal part is of constant coefficients, whilethe energy level (H1(Q)) terms may be variable in both space and time, andof low regularity L1(Q). Verifiable geometric conditions are imposed on theunobserved portion г0. Then: we first establish a Carleman-type inequalityfor H1; 1(Q)-solutions of the hyperbolic equation with no interior lower-orderterms. From here, we deduce global uniqueness results for H1;1(Q) solutionsof the hyperbolic equation satisfying Neumann B.C. on all of г, and DirichletB.C. on г0, over a time T greater than an explicit time T0. T0 is optimalif, e.g., г0 is flat. Finally we obtain continuous observability (or stabilization)inequalities with an explicit constant. A three-part appendix, of purelygeometric nature, provides several independent approaches leading to variousgeneral classes of triples f­; 0; 1g which satisfy the geometric conditions ofSection 1, and, more relevantly, the geometric conditions of the far more generalSection 10: see Theorem C.1 in Appendix C. In particular: 0 may be flat; 0 may be either convex or concave; ¡0 may be logarithm convex or concave;etc. In the case of a disk, we can take the unobserved part ¡0 to be as closeto a half-circumference as we please: an indication that our results in Section10 are sharp. Finally, in line with the AMS Conference at the University ofColorado, we point out throughout some open geometric questions, as well assome potential extensions which would require geometric methods. Extensionof the fundamental Lemma 3.1 to the case of variable (in space) coefficientsof the principal part has already been accomplished [L-T-Y-Z.1] by means ofBochner's techniques in Riemann geometry, in the style of [L-T-Y.1–3], [Y.1].