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The classical theory of the geometrical string is developed as the theory of a simple, surface-forming timelike bivector field in an arbitrary background space-time. The stress-energy tensor for a perfect dust o f such strings is written down, and the conservation laws for such a dust, as well as the equations of motion of the string, are derived from the vanishing of the divergence of the stress-energy tensor. (The boundary conditions for the open string are also derived from the junction conditions for the stress-energy tensor in Appendix A.) The generalization of this model to null strings, and to a perfect fluid of strings, are discussed, and will form the subject o f the second and third papers in this series. The problem of a fully general-relativistic string theory, and an alternate approach to the string, based upon defining an acceleration tensor for two-(and higher) dimensional subspaces, are also discussed. I . INTRODUCTIONIn recent y e a r s there has been extensive discussion of the theory of the geometrical string, primarily because of i t s application a s a possible interpretation of the dual resonance model of elementary particles.' However, the theory has also been discussed on the classical level2; and, indeed, f r o m this point of view it provides a r emarkably natural generalization of the relativistic theory of a structureless point particle. This generalization could have been made a t any time after 1905, and it is r a t h e r surprising that such a natural structure should not have been investigated much earlier. This paper will be exclusively concerned with the classical theory of the string; the hope of generalizing special-relativistic string theory into a fully general-relativistic one provides i t s ultimate m~t i v a t i o n .~The geometrical string will b e treated a s the theory of a surface-forming simple bivector field, subject to field equations which determine the s u rface. This bivector field i s usually treated mathematically by introducing a parametrization of the surface, and a pair of linearly independent vectors, derived f r o m the parametrization, which span the surface. While t h e r e i s nothing wrong with this procedure, it may tend to make one lose sight of the fact that the resulting theory must b e invariant under the family of possible reparametrizations of the surface which leave the bivector unaltered. However, it is equally simple to t r e a t the bivector field intrinsically, a s we shall demons t r a t e , and perhaps more natural t o do so, in the sense that it is m o r e natural to discuss any geometrical figure intrinsically r a t h e r than via i t s components with respect to some basis.The next section will review some well-known mathematical results on simple surface-forming bivector fields that will be needed in the following In Sec. 111, we shall review the "thickening" of a point particle provided by the incoherent dust model of matter, and r e c a l l how the equations of motion and conservation law for the dust may b e derived f r o m the van...

It is well known that gravitational fields may be locally the same but globally distinct due to differences in the topology of their underlying manifolds. Globally stationary but locally static gravitational fields provide an example of gravitational fields which are locally the same but globally distinct in spite of the homeomorphism of their underlying manifolds. Any static metric on a space-time manifold with nonvanishing first Betti number R , is shown to generate an R ,-parameter family of such solutions. These fields are seen to provide a gravitational analog of the electromagnetic Aharonov-Bohm effect. The exterior field of a rotating infinite cylinder of matter is discussed as an exactly soluble example.