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$%P. Jaworski, On Witt rings of function fields of real analytic surfaces and curves, Rev. Mat. Univ. Complut. Madrid 10 (1997), supplementary, 153–171. %$

Let be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold of the same dimension. Moreover by a prime divisor of we shall mean the irreducible germ along of a codimension one subvariety of which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification . In the one-dimensional case prime divisors are just points, in the two-dimensional analytic curves or elliptic points (intersections of two conjugated complex analytic curves). Every such divisor induces a discrete valuation on the field of meromorphic functions on --the order of the zero or minus the order of the pole of the function. Therefore it induces the so called residue homomorphisms (first and second) of the Witt group of the field to the Witt group of the residue field --the function field of the divisor.

The main goal of this paper is to show that the intersection of kernels of all second residue homomorphisms associated to prime divisors is isomorphic to the Witt group of the Riemannian bundles on .

As an example of an application of this result we provide the new proof of the Artin-Lang property for one and two dimensional real analytic manifolds (both compact and noncompact), which is neither based on the description of all possible orderings of the field of meromorphic functions nor on the compactification of the variety.

1991 Mathematics Subject Classification: 14P15, 11E81, 12D15.