Total torsion of three-dimensional lines of curvature

A curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ lies on an oriented hypersurface S of M, we say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ is three-dimensional and closed. We show that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ is a well-positioned line of curvature of S, then its total torsion is an integer multiple of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\pi $$\end{document}2π; and that, conversely, if the total torsion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ is an integer multiple of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\pi $$\end{document}2π, then there exists an oriented hypersurface of M in which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}γ vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.


Introduction and main result
In classical differential geometry, the total torsion theorem states that the total torsion of a closed spherical curve vanishes; see [3,10,11,4] and [7, p. 170].
Closely related to Theorem 1.1 is the following result of Qin and Li.
Theorem 1.2 ( [9]).Let S be a (smooth) oriented surface in R 3 .If γ is a closed line of curvature of S, then the total torsion is an integer multiple of 2π.Conversely, if the total torsion of a closed curve in R 3 is an integer multiple of 2π, then it can appear as a line of curvature of an oriented surface.
Theorem 1.1 and the first part of Theorem 1.2 have been generalized to threedimensional orientable Riemannian manifolds of constant curvature M 3 c [8]; see also [2,15] for related results.In the present note we shall see that, under suitable assumptions, both theorems remain valid when M 3 c is replaced by an arbitrary Riemannian manifold M m ≡ M , provided one restricts the attention to threedimensional curves; roughly speaking, a curve in M is three-dimensional if it has one curvature and one "torsion", all other curvature functions being zero.As we explain below, in that case one should interpret "torsion" as a signed version of Spivak's "second curvature function" [13, p. 22].
Let γ be a unit-speed curve I → M , let N be a unit normal vector field along γ, and let π H be the orthogonal projection onto H = (γ ′ ⊕ N ) ⊥ .We say that N is torsion-defining if there exists a smooth unit vector field W (N ) along γ that is everywhere parallel to T g = −π H D t N .If N is torsion-defining, then the function τ g = ⟨T g , W (N )⟩ is called the (first) geodesic torsion of γ with respect to N .In particular, if D t γ ′ is never zero, then the geodesic torsion of γ with respect to the principal normal P = D t γ ′ /κ is called the (first) torsion of γ, and γ is said to be a Frenet curve.
The logic behind our terminology is the following.In the same way a generic curve in R 3 has one (unsigned) curvature plus one (signed) torsion, a generic curve in M may have one (unsigned) curvature plus m − 2 (signed) torsions; cf.[13].On the other hand, since we never deal with higher-order torsions, we typically speak of "torsion" as a shorthand for "first torsion".Now, to state our generalization of Theorems 1.1 and 1.2, let S be an oriented hypersurface of M , and let N S be its unit normal.A Frenet curve on S is said to be well positioned if N S , P , and W (P ) are everywhere coplanar.
Theorem 1.3.Suppose that γ is three-dimensional, i.e., that γ is a Frenet curve such that W (P ) is parallel in H(P ); see Definition 6.1.If γ is a well-positioned closed line of curvature of S, then the total torsion of γ is an integer multiple of 2π; in particular, the total torsion vanishes when S is convex, i.e., when the second fundamental form of S is positive definite.Conversely, if γ is open, then there exists an orientable hypersurface in which γ is a well-positioned line of curvature; if γ is closed, then the same holds provided the total torsion of γ is an integer multiple of 2π.
Clearly, when dim M = 3, every Frenet curve is three-dimensional, and every Frenet curve on S is well positioned.Specializing the theorem to that case, we may state the following result.
Corollary 1.4.Suppose that dim M = 3 and that γ is a closed Frenet curve.If γ is a line of curvature of S, then the total torsion of γ is an integer multiple of 2π; in particular, the total torsion vanishes when S is convex.Conversely, if the total torsion of γ is an integer multiple of 2π, then there exists an orientable surface in which γ is a line of curvature.Remark 1.5.If dim M = 3, then every regular curve with nonvanishing curvature is a Frenet curve.
We will obtain Theorem 1.3 as a corollary of a more general statement involving the geodesic torsion of γ with respect to an arbitrary unit normal vector field N along γ, in which the assumption that γ is three-dimensional is replaced by the condition that N is a parallel rotation of N S .
Let N and Z be unit normal vector fields along γ.We say that Z is a rotation of N if there exists a continuous unit vector field (2) H, N , and Z are everywhere linearly dependent.Clearly, if N ∧ Z is nowhere zero, then the vector field H is defined up to a sign.Now, suppose that Z is a rotation of N .Then we can write for some continuous function θ : I → R.
Definition 1.6.A rotation of N is said to be parallel if H is parallel with respect to the induced connection on H, and closed if θ(ℓ) − θ(0) = 2nπ for some n ∈ Z.
(1) If dim M = 3, then any unit normal vector field along γ is a parallel rotation of N .(2) If γ is closed, then so is any rotation of N .
Theorem 1.8.If γ is a line of curvature of S, then the total geodesic torsion of γ with respect to any closed parallel rotation of N S is an integer multiple of 2π.Conversely, suppose that N is torsion-defining and that W (N ) is parallel in H.If γ is open, then there exists an orientable hypersurface of M in which γ is a line of curvature; if γ is closed, then the same holds provided the total geodesic torsion of γ with respect to N is an integer multiple of 2π.
Remark 1.9.It follows from section 4 that if γ is a line of curvature of S and P is a parallel rotation of N S , then γ is three-dimensional.
The remainder of the paper is organized a follows.In section 2 we set up some notations.In section 3 we generalize the well-known concepts of geodesic curvature, normal curvature, and geodesic torsion of a curve on a surface in R 3 to a curve on a hypersurface of M ; although, under reasonable assumptions, one may define m − 2 geodesic curvatures and geodesic torsions, for the sake of simplicity we shall limit ourselves to first-order curvatures.In section 4 we obtain formulas expressing the curvature vectors of γ with respect to a rotation of N in terms of the rotation angle.Finally, in sections 5 and 6 we prove Theorems 1.8 and 1.3, respectively.

Preliminaries
In this section we discuss some preliminaries.Let M be an m-dimensional Riemannian manifold, let γ be a smooth unit-speed curve I → M , and let T M | γ be the ambient tangent bundle over γ.Recall that We define a distribution of rank r along γ to be a rank-r subbundle of T M | γ .
Let D be a distribution of rank r along γ, and let D ⊥ be the distribution of rank m − r along γ whose fiber at t is the orthogonal complement D ⊥ t of D t in T γ(t) M , so that T M | γ splits as accordingly, we write X = X v + X h for any vector field X along γ.
In this setting, the tangential projection is the map

Darboux curvatures and curvature vectors
The purpose of this section is to extend the classical notions of geodesic curvature, normal curvature, and geodesic torsion of a curve on a surface in R 3 to a curve on a hypersurface of M .
Let γ be a (smooth) unit-speed curve I → M , let N be a unit normal vector field along γ, and let H(N ) ≡ H be the distribution of rank m − 2 along γ whose fiber at t is the orthogonal complement of E(t) = γ ′ (t) and N (t) in T γ(t) M .Denoting by D t the covariant derivative along γ, we define • the (first) geodesic curvature vector K g of γ with respect to N by • the normal curvature vector K n of γ with respect to N by where N = span N ; • the (first) geodesic torsion vector T g of γ with respect to N by To express these vector fields in coordinates, let (H 1 , . . ., H m−2 ) be a smooth orthonormal frame for H. Then there are functions κ 1 g , . . ., κ m−2 g , κ n , and τ 1 g , . . ., τ m−2 g such that Note that, since (E, H 1 , . . ., H m−2 , N ) is orthonormal, the following equations hold for all j = 1, . . ., m − 2: The curvature vectors allow us to define corresponding curvature functions.In one case, the definition is trivial: the function κ n = ⟨D t E, N ⟩ is called the normal curvature of γ with respect to N .For the remaining two cases, we proceed as follows.
We say that N is curvature-defining if there exists a smooth unit vector field V (N ) ≡ V along γ that is everywhere parallel to K g .If N is curvature-defining, then the function κ g = ⟨K g , V ⟩ is called the (first) geodesic curvature of γ with respect to N .
Similarly, we say that N is torsion-defining if there exists a smooth unit vector field W (N ) ≡ W along γ that is everywhere parallel to T g .If N is torsion-defining, then the function τ g = ⟨T g , W ⟩ is called the (first) geodesic torsion of γ with respect to N .
It is clear that both κ g and τ g are defined up to a sign.Armed with the notion of geodesic torsion, we may now define torsion.Suppose that the curvature κ = ∥D t E∥ of γ is nowhere zero, so that the principal normal P = D t E/κ is well-defined.The geodesic torsion vector of γ with respect to P is called the (first) torsion vector of γ.In particular, if P is torsion-defining, then the geodesic torsion of γ with respect to P is called the (first) torsion of γ.
Remark 3.1.If P is well-defined, then the normal curvature of γ with respect to P coincides with the curvature of γ, while the geodesic curvature with respect to P vanishes.
To see that our curvature functions naturally extend the classical Darboux curvatures, consider an oriented hypersurface S of M , and let N S be its unit normal.If γ is a curve on S, then the geodesic (resp., normal) curvature vector of γ (with respect to N S ) is the projection onto T S (resp., N S) of the ambient acceleration D t E of γ; and if γ is not a geodesic of M , then the geodesic torsion vector of γ at γ(t) is nothing but the torsion vector of the S-geodesic passing from γ(t) with tangent vector γ ′ (t) [14, p. 193].
Yet another indication of the naturality of our definition of geodesic torsion is provided by the following lemma, which will play a key role in the proof of Theorem 1.8.Lemma 3.2.A curve on S is a line of curvature if and only if its geodesic torsion vector with respect to N S vanishes.Remark 3.3.Under suitable assumptions, one may define m − 2 geodesic curvature and (geodesic) torsion functions.For instance, the second geodesic torsion is defined as follows.Let H 2 = (T ⊕ N ⊕ T g ) ⊥ , let π H 2 be the orthogonal projection onto H 2 , and let If T g is itself torsion-defining, i.e., there exists a smooth unit vector field W 2 along γ that is everywhere parallel to T g,2 , then the function τ g,2 = ⟨T g,2 , W 2 ⟩ is called the second geodesic torsion of γ with respect to N .Higher-order geodesic torsions are defined similarly.

Rotating the normal
Suppose that the normal vector N along γ rotates about the curve's tangent.Then how do the curvature vectors change?The purpose of this section is to answer such question.
Let Z be a rotation of N .Then, by definition, there exists a unit normal vector field H(N, Z) ≡ H ∈ Γ(H) along γ such that N , Z, and H are everywhere linearly dependent; besides, there is a continuous function θ : Denoting Z by N (θ), we call the function θ the rotation angle of N (θ) with respect to H. Now, let (H 1 , . . ., H m−2 ) be a smooth orthonormal frame for H = (E ⊕ N ) ⊥ , with H 1 = H.It follows that N (θ) = − sin(θ)H 1 + cos(θ)N, while the vector fields . . .
Lemma 4.1.The curvature vectors of γ with respect to N (θ) are given by where µ j = ⟨D t H j , H 1 ⟩, and where c and s are shorthands for cos(θ) and sin(θ), respectively.

Proof of Theorem 1.8
Here we prove our most general result, Theorem 1.8 in the introduction.
To begin with, suppose that γ is a line of curvature of S and that N S (θ) is a parallel rotation of N S .Then the geodesic torsion of γ with respect to N S vanishes and the vector field H(N S , N S (θ)) is parallel in H.

Three-dimensional curves
Let γ : I → M be a Frenet curve, let H 1 = W (P ), and let (H 2 , . . ., H m−2 ) be a parallel frame for the orthogonal complement of H 1 in H(P ).Definition 6.1.We say that γ is three-dimensional if the following equations hold: It is clear that γ is three-dimensional if and only if W (P ) is parallel in H(P ).The purpose of this section is to prove Theorem 1.3 in the introduction.
Proof of Theorem 1.3.Suppose that P is a parallel rotation of N S , and let θ be the rotation angle of P with respect to W (P ).We know from the proof of Theorem 1.8 that if γ is a line of curvature and P is a closed rotation, then On the other hand, applying Lemma 4.1, we observe that the normal curvature of γ with respect to N S is related to the curvature κ by the relation Suppose that M is convex, so that κ n > 0. Since κ > 0, we have cos(θ) > 0, from which we conclude that θ(ℓ) − θ(0) ∈ (−π, π).