In  () M. Bartolozzi and Aldo Bressan considered a regular portion Δσ of a m-dim. Riemaniann manifold σ=σ(t), possibly moving in a Euclidean space Sv and introduced the notion of (volume preserving) velocity fields as rigid as possible—in short V F R P (V P V F R P)—as an extension of rigid velocity fields —R V F—of classical physics. Existence problems were not considered. In the present paper, written in two parts, we observe that, for V F R P and V P V F R P, and existence theorem similiar to the one for R V F does not hold. Therefore, even if some V F R P (V P V R F P) do always exist, under vevy special constraints, it is natural to look for an alternative definition for the classes V F R P (V P V F R P), so that the existence question has a positive answer in a fully general case. To this end, here in Part 1, a new class V F R P of velocity fields as rigid as possible subject to a set of linear and continuous constraints Γ is introduced. For this class a general existence theorem is proven. A uniqueness result, up to an intrinsecally rigid vector field, is also provided. In part 2 we show how the set of constraints can be expressed in various ways, using integral conditions on volumes or hypersurfaces, essentially equivalent to prescribing the value of the field vi and of its spin vi/h at some point P0. We do this first using a generic field Σ(x)=(e1 (x), ..., e(m) (x)) of orthonormal m-tuples of tangent vectors on Δσ. Phisically interesting choices of Σ(x) are then considerd: the ones which are as euclidan (i.e. as parallel) as possible. The existence of such Σ(x) is proven. It is also shown that, in the two-dimensional case, any two fields Σ, Σ’, both as euclidean as possible, differ by a constant rotation.
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