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We use the solution set of a real ordinary differential equation which has order n ≥ 2 to construct a smooth curve σ in R n . We describe when σ is a proper embedding of infinite length with finite total first curvature. Statement of resultsLet σ be an immersion of R into R n for n ≥ 2; if σ is proper, then the length of σ is then necessarily infinite. The first curvature κ and the total first curvature κ[σ] are given, respectively, by: κ := ||σ ∧σ|| ||σ|| 3 and κ[σ] := ∞ −∞ κ||σ||dx . (1.a) 1.1. History. Fenchel [16] showed that a simple closed curve in R 3 had κ[σ] ≥ 2π. Fáry [15] and Milnor [18] showed the total curvature of any knot is greater than 4π. Castrillón López and Fernández Mateos [6], and Kondo and Tanaka [17] have examined the global properties of the total curvature of a curve in an arbitrary Riemannian manifold. The total curvature of open plane curves of fixed length in R 2 was studied by Enomoto [11]. The analogous question for S 2 was examined by Enomoto and Itoh [12, 13]. Enomoto, Itoh, and Sinclair [14] examined curves in R 3 . We also refer to related work of Sullivan [19]. Borisenko and Tenenblat [4] studied the problem in Minkowski space. Buck and Simon [5] and Diao and Ernst [9] studied this invariant in the context of knot theory, and Ekholm [10] used this invariant in the context of algebraic topology. Alexander, Bishop, and Ghrist [1]extended these notions to more general spaces than smooth manifolds. The total curvature also appears in the study of Plateau's problem -see the discussion in Desideri and Jakob [8].The literature on the subject is a vast one and we have only cited a few representative papers to give a flavor for the subject.The papers cited above focused on closed curves, polygonal curves, knotted curves, curves with fixed endpoints, curves with finite length and pursuit curves. This present paper deals, by contrast, with properly embedded curves which arise from ordinary differential equations. We take as our starting point a real constant coefficient ordinary differential operator P of degree n ≥ 2 of the form: P (y) := y (n) + c n−1 y (n−1) + · · · + c 0 y .2010 Mathematics Subject Classification. 53A04.

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