1 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, 4, 10, 11] and [7, p. 170].

Theorem 1.1

Let \(I= [0, \ell ]\), and let \(\gamma :I \rightarrow \mathbb {R}^{3}\) be a smooth regular curve. If \(\gamma \) is closed and \(\gamma (I)\in \mathbb {S}^{2}\), then

$$\begin{aligned} \int _{0}^{\ell } \tau \,dt = 0. \end{aligned}$$

Theorem 1.1 manifests the fact that “the torsion of a closed curve lying on a surface in \(\mathbb {R}^{3}\) is somehow constrained by the geometry of [the] surface” [8, p. 111]; see, e.g., [1, 5, 6, 12] for further evidence of the same fact.

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 \(\mathbb {R}^{3}\). If \(\gamma \) is a closed line of curvature of S, then the total torsion is an integer multiple of \(2\pi \). Conversely, if the total torsion of a closed curve in \(\mathbb {R}^{3}\) is an integer multiple of \(2\pi \), 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 three-dimensional orientable Riemannian manifolds of constant curvature \(M_{c}^{3}\) [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 \(M^{m}\,{\equiv }\,M\), provided one restricts the attention to three-dimensional 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 \(\gamma \) be a unit-speed curve \(I \rightarrow M\), let N be a unit normal vector field along \(\gamma \), and let \(\pi _{\mathcal {H}}\) be the orthogonal projection onto \(\mathcal {H} = (\gamma ' \oplus N)^{\perp }\). We say that N is torsion-defining if there exists a smooth unit vector field W(N) along \(\gamma \) that is everywhere parallel to \(T_{g} = -\pi _{\mathcal {H}}D_{t}N\). If N is torsion-defining, then the function \(\tau _{g} = \langle T_{g}, W(N)\rangle \) is called the (first) geodesic torsion of \(\gamma \) with respect to N. In particular, if \(D_{t}\gamma '\) is never zero, then the geodesic torsion of \(\gamma \) with respect to the principal normal \(P = D_{t}\gamma '/\kappa \) is called the (first) torsion of \(\gamma \), and \(\gamma \) is said to be a Frenet curve.

The logic behind our terminology is the following. In the same way a generic curve in \(\mathbb {R}^{3}\) 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 \(\gamma \) is three-dimensional, i.e., that \(\gamma \) is a Frenet curve such that W(P) is parallel in \(\mathcal {H}(P)\); see Definition 6.1. If \(\gamma \) is a well-positioned closed line of curvature of S, then the total torsion of \(\gamma \) is an integer multiple of \(2\pi \); in particular, the total torsion vanishes when S is convex, i.e., when the second fundamental form of S is positive definite. Conversely, if \(\gamma \) is open, then there exists an orientable hypersurface in which \(\gamma \) is a well-positioned line of curvature; if \(\gamma \) is closed, then the same holds provided the total torsion of \(\gamma \) is an integer multiple of \(2\pi \).

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 \(\gamma \) is a closed Frenet curve. If \(\gamma \) is a line of curvature of S, then the total torsion of \(\gamma \) is an integer multiple of \(2\pi \); in particular, the total torsion vanishes when S is convex. Conversely, if the total torsion of \(\gamma \) is an integer multiple of \(2\pi \), then there exists an orientable surface in which \(\gamma \) 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 \(\gamma \) with respect to an arbitrary unit normal vector field N along \(\gamma \), in which the assumption that \(\gamma \) 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 \(\gamma \). We say that Z is a rotation of N if there exists a continuous unit vector field \(H(N, Z) \equiv H\) such that

  1. (1)

    \(\langle H, \gamma ' \rangle = \langle H, N \rangle = 0\), i.e., \(H \in \Gamma (\mathcal {H})\);

  2. (2)

    H, N, and Z are everywhere linearly dependent.

Clearly, if \(N \wedge Z\) 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

$$\begin{aligned} Z \equiv N(\theta )= - \sin (\theta ) H + \cos (\theta ) N\ \end{aligned}$$

for some continuous function \(\theta :I \rightarrow \mathbb {R}\).

Definition 1.6

A rotation of N is said to be parallel if H is parallel with respect to the induced connection on \(\mathcal {H}\), and closed if \(\theta (\ell )-\theta (0) = 2n\pi \) for some \(n \in \mathbb {Z}\).

Remark 1.7

  1. (1)

    If \(\dim M = 3\), then any unit normal vector field along \(\gamma \) is a parallel rotation of N.

  2. (2)

    If \(\gamma \) is closed, then so is any rotation of N.

Theorem 1.8

If \(\gamma \) is a line of curvature of S, then the total geodesic torsion of \(\gamma \) with respect to any closed parallel rotation of \(N_{S}\) is an integer multiple of \(2\pi \). Conversely, suppose that N is torsion-defining and that W(N) is parallel in \(\mathcal {H}\). If \(\gamma \) is open, then there exists an orientable hypersurface of M in which \(\gamma \) is a line of curvature; if \(\gamma \) is closed, then the same holds provided the total geodesic torsion of \(\gamma \) with respect to N is an integer multiple of \(2\pi \).

Remark 1.9

It follows from Sect. 4 that, if \(\gamma \) is a line of curvature of S and P is a parallel rotation of \(N_{S}\), then \(\gamma \) is three-dimensional.

The remainder of the paper is organized a follows. In Sect. 2 we set up some notations. In Sect. 3 we generalize the well-known concepts of geodesic curvature, normal curvature, and geodesic torsion of a curve on a surface in \(\mathbb {R}^{3}\) 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 Sect. 4 we obtain formulas expressing the curvature vectors of \(\gamma \) with respect to a rotation of N in terms of the rotation angle. Finally, in Sects. 5 and 6 we prove Theorems 1.8 and 1.3, respectively.

2 Preliminaries

In this section we discuss some preliminaries.

Let M be an m-dimensional Riemannian manifold, let \(\gamma \) be a smooth unit-speed curve \(I \rightarrow M\), and let \(TM |_{\gamma }\) be the ambient tangent bundle over \(\gamma \). Recall that

$$\begin{aligned} TM|_{\gamma } = \bigsqcup _{t \in I} T_{\gamma (t)}M. \end{aligned}$$

We define a distribution of rank r along \(\gamma \) to be a rank-r subbundle of \(TM |_{\gamma }\).

Let \(\mathcal {D}\) be a distribution of rank r along \(\gamma \), and let \(\mathcal {D}^{\perp }\) be the distribution of rank \(m-r\) along \(\gamma \) whose fiber at t is the orthogonal complement \(\mathcal {D}_{t}^{\perp }\) of \(\mathcal {D}_{t}\) in \(T_{\gamma (t)}M\), so that \(TM|_{\gamma }\) splits as

$$\begin{aligned} TM|_{\gamma } = \mathcal {D} \oplus \mathcal {D}^{\perp }; \end{aligned}$$

accordingly, we write

$$\begin{aligned} X = X^{v} + X^{h} \end{aligned}$$

for any vector field X along \(\gamma \).

In this setting, the tangential projection is the map \(\pi _{\mathcal {D}} :\Gamma (TM|_{\gamma }) \rightarrow \Gamma (\mathcal {D})\) given by

$$\begin{aligned} X \mapsto X^{v}. \end{aligned}$$

Likewise, the normal projection is the map \(\pi ^{\perp }_{\mathcal {D}} :\Gamma (TM|_{\gamma }) \rightarrow \Gamma (\mathcal {D}^{\perp })\) sending each X to the corresponding \(X^{h}\).

3 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 \(\mathbb {R}^{3}\) to a curve on a hypersurface of M.

Let \(\gamma \) be a (smooth) unit-speed curve \(I \rightarrow M\), let N be a unit normal vector field along \(\gamma \), and let \(\mathcal {H}(N) \equiv \mathcal {H}\) be the distribution of rank \(m-2\) along \(\gamma \) whose fiber at t is the orthogonal complement of \(E(t) = \gamma '(t)\) and N(t) in \(T_{\gamma (t)}M\). Denoting by \(D_{t}\) the covariant derivative along \(\gamma \), we define

  • the (first) geodesic curvature vector \(K_{g}\) of \(\gamma \) with respect to N by

    $$\begin{aligned} K_{g} = \pi _{\mathcal {H}} D_{t}E; \end{aligned}$$

  • the normal curvature vector \(K_{n}\) of \(\gamma \) with respect to N by

    $$\begin{aligned} K_{n} = \pi _{\mathcal {N}} D_{t}E, \end{aligned}$$

    where \(\mathcal {N} = {{\,\textrm{span}\,}}N\);

  • the (first) geodesic torsion vector \(T_{g}\) of \(\gamma \) with respect to N by

    $$\begin{aligned} T_{g} = -\pi _{\mathcal {H}} D_{t}N. \end{aligned}$$

To express these vector fields in coordinates, let \((H_{1}, \dotsc , H_{m-2})\) be a smooth orthonormal frame for \(\mathcal {H}\). Then there are functions \(\kappa _{g}^{1}, \dotsc , \kappa _{g}^{m-2}\), \(\kappa _{n}\), and \(\tau _{g}^{1}, \dotsc , \tau _{g}^{m-2}\) such that

$$\begin{aligned} K_{g}&= \kappa _{g}^{1}H_{1} + \cdots + \kappa _{g}^{m-2} H_{m-2},\\ K_{n}&= \kappa _{n} N,\\ T_{g}&= \tau _{g}^{1}H_{1} + \cdots + \tau _{g}^{m-2} H_{m-2}. \end{aligned}$$

Note that, since \((E, H_{1}, \dotsc , H_{m-2}, N)\) is orthonormal, the following equations hold for all \(j =1, \dotsc , m-2\):

$$\begin{aligned}&D_{t}E = \kappa _{g}^{1}H_{1} + \cdots + \kappa _{g}^{m-2}H_{m-2} + \kappa _{n}N, \\&D_{t}H_{j} = -\kappa _{g}^{j}E + \tau _{g}^{j}N + \pi _{\mathcal {H}} D_{t}H_{j}, \\&D_{t}N = -\kappa _{n}E - \tau _{g}^{1}H_{1} - \cdots - \tau _{g}^{m-2} H_{m-2}. \end{aligned}$$

The curvature vectors allow us to define corresponding curvature functions. In one case, the definition is trivial: the function \(\kappa _{n} = \langle D_{t}E, N \rangle \) is called the normal curvature of \(\gamma \) 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 \(V(N) \equiv V\) along \(\gamma \) that is everywhere parallel to \(K_{g}\). If N is curvature-defining, then the function \(\kappa _{g} = \langle K_{g}, V \rangle \) is called the (first) geodesic curvature of \(\gamma \) with respect to N.

Similarly, we say that N is torsion-defining if there exists a smooth unit vector field \(W(N) \equiv W\) along \(\gamma \) that is everywhere parallel to \(T_{g}\). If N is torsion-defining, then the function \(\tau _{g} = \langle T_{g}, W \rangle \) is called the (first) geodesic torsion of \(\gamma \) with respect to N.

It is clear that both \(\kappa _{g}\) and \(\tau _{g}\) are defined up to a sign.

Armed with the notion of geodesic torsion, we may now define torsion. Suppose that the curvature \(\kappa = \Vert D_{t}E \Vert \) of \(\gamma \) is nowhere zero, so that the principal normal \(P = D_{t}E/\kappa \) is well-defined. The geodesic torsion vector of \(\gamma \) with respect to P is called the (first) torsion vector of \(\gamma \). In particular, if P is torsion-defining, then the geodesic torsion of \(\gamma \) with respect to P is called the (first) torsion of \(\gamma \).

Remark 3.1

If P is well-defined, then the normal curvature of \(\gamma \) with respect to P coincides with the curvature of \(\gamma \), 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 \(\gamma \) is a curve on S, then the geodesic (resp., normal) curvature vector of \(\gamma \) (with respect to \(N_{S}\)) is the projection onto TS (resp., NS) of the ambient acceleration \(D_{t}E\) of \(\gamma \); and if \(\gamma \) is not a geodesic of M, then the geodesic torsion vector of \(\gamma \) at \(\gamma (t)\) is nothing but the torsion vector of the S-geodesic passing from \(\gamma (t)\) with tangent vector \(\gamma '(t)\) [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 \(\mathcal {H}_{2} = (T \oplus N \oplus T_{g})^{\perp }\), let \(\pi _{\mathcal {H}_{2}}\) be the orthogonal projection onto \(\mathcal {H}_{2}\), and let

$$\begin{aligned} T_{g,2} = -\pi _{\mathcal {H}_{2}} D_{t}T_{g}. \end{aligned}$$

If \(T_{g}\) is itself torsion-defining, i.e., there exists a smooth unit vector field \(W_{2}\) along \(\gamma \) that is everywhere parallel to \(T_{g,2}\), then the function \(\tau _{g,2} = \langle T_{g,2}, W_{2} \rangle \) is called the second geodesic torsion of \(\gamma \) with respect to N. Higher-order geodesic torsions are defined similarly.

4 Rotating the normal

Suppose that the normal vector N along \(\gamma \) 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 \(H(N,Z) \equiv H \in \Gamma (\mathcal {H})\) along \(\gamma \) such that N, Z, and H are everywhere linearly dependent; besides, there is a continuous function \(\theta :I \rightarrow \mathbb {R}\) such that

$$\begin{aligned} Z = -\sin (\theta )H + \cos (\theta )N. \end{aligned}$$

Denoting Z by \(N(\theta )\), we call the function \(\theta \) the rotation angle of \(N(\theta )\) with respect to H.

Now, let \((H_{1}, \dotsc , H_{m-2})\) be a smooth orthonormal frame for \(\mathcal {H} = (E \oplus N)^{\perp }\), with \(H_{1} = H\). It follows that

$$\begin{aligned} N(\theta ) = -\sin (\theta ) H_{1} + \cos (\theta ) N, \end{aligned}$$

while the vector fields

$$\begin{aligned} H_{1}(\theta )&= \cos (\theta )H_{1} + \sin (\theta ) N,\\ H_{2}(\theta )&= H_{2},\\&\quad {\vdots } \\ H_{m-2}(\theta )&= H_{m-2} \end{aligned}$$

span \(\mathcal {H}(N(\theta )) = (E \oplus N(\theta ))^{\perp }\).

Lemma 4.1

The curvature vectors of \(\gamma \) with respect to \(N(\theta )\) are given by

$$\begin{aligned} K_{g}(\theta )&= \left( \kappa _{g}^{1} c + \kappa _{n} s \right) H_{1}(\theta ) + \kappa _{g}^{2} H_{2}(\theta ) + \cdots + \kappa _{g}^{m-2} H_{m-2}(\theta ),\\ K_{n}(\theta )&= \left( -\kappa _{g}^{1} s + \kappa _{n} c \right) N(\theta ),\\ T_{g}(\theta )&= \left( \theta ' + \tau _{g}^{1} \right) H_{1}(\theta ) + \left( \tau _{g}^{2} c - \mu _{2} s \right) H_{2}(\theta ) + \cdots + \left( \tau _{g}^{m-2} c - \mu _{m-2} s \right) H_{m-2}(\theta ), \end{aligned}$$

where \(\mu _{j} = \langle D_{t} H_{j}, H_{1} \rangle \), and where c and s are shorthands for \(\cos (\theta )\) and \(\sin (\theta )\), respectively.

5 Proof of Theorem 1.8

Here we prove our most general result, Theorem 1.8 in the introduction.

To begin with, suppose that \(\gamma \) is a line of curvature of S and that \(N_{S}(\theta )\) is a parallel rotation of \(N_{S}\). Then the geodesic torsion of \(\gamma \) with respect to \(N_{S}\) vanishes and the vector field \(H(N_{S}, N_{S}(\theta ))\) is parallel in \(\mathcal {H}\).

Let \((H_{1}, \dotsc , H_{m-2})\) be a smooth orthonormal frame for \((E \oplus N_{S})^{\perp }\) such that \(H_{1} = H\). Applying Lemma 4.1, we deduce that \(T_{g}(\theta ) = \theta ' H_{1}(\theta )\), which implies that \(N_{S}(\theta )\) is torsion-defining and that \(\theta ' = \pm \tau _{g}(\theta )\).

Since

$$\begin{aligned} \int _{0}^{\ell } \theta ' \, dt = \theta (\ell ) - \theta (0), \end{aligned}$$

it follows that, when \(N_{S}(\theta )\) is a closed rotation of \(N_{S}\),

$$\begin{aligned} \int _{0}^{\ell } \tau _{g}(\theta ) = 2n\pi \quad \hbox { for some}\ n \in \mathbb {Z}, \end{aligned}$$

as desired.

Conversely, given any (torsion-defining) unit normal vector field N along \(\gamma \), suppose that \(W(N) \equiv W\) is parallel in \(\mathcal {H}\). Choose an orthonormal frame \((H_{1}, \dotsc , H_{m-2})\) for \(\mathcal {H}\), with \(H_{1} = W\), and let

$$\begin{aligned} N(\theta )&= -\sin (\theta )H_{1} + \cos (\theta )N,\\ H_{1}(\theta )&= \cos (\theta )H_{1} + \sin (\theta )N, \end{aligned}$$

where

$$\begin{aligned} \theta (t) = -\int _{0}^{t} \tau _{g}(s) \, ds. \end{aligned}$$
(1)

(Note that \(N(\theta )\) is a parallel rotation of N.)

Define a map \(\sigma :[0, \ell ] \times \mathbb {R}^{m-1} \rightarrow M\) by

$$\begin{aligned} \sigma (t, u) = \exp _{\gamma (t)}\left( u^{1}H_{1}(\theta )(t) + u^{2}H_{2}(t) + \cdots + u^{m-2} H_{m-2}(t)\right) . \end{aligned}$$

It is clear that \(\sigma \) is a smooth immersion in a neighborhood of \([0, \ell ] \times \{0 \}\); besides, its image is normal to \(N(\theta )\) along \(\gamma \).

It remains to show that \(\gamma \) is a line of curvature of \(\sigma \), i.e., that the geodesic torsion \(\tau _{g}(\theta )\) of \(\gamma \) with respect to \(N(\theta )\) vanishes. Differentiating (1), we have

$$\begin{aligned} \theta ' = -\tau _{g}^{1}, \end{aligned}$$

which implies \(\tau _{g}^{1}(\theta ) = 0\), as desired. Since \(\tau _{g}^{2} = \cdots = \tau _{g}^{m-1} = 0\) and \(H_{1}\) is parallel in \(\mathcal {H}\), we conclude that \(\tau _{g}(\theta ) = 0\) by Lemma 4.1.

6 Three-dimensional curves

Let \(\gamma :I \rightarrow M\) be a Frenet curve, let \(H_{1} = W(P)\), and let \((H_{2}, \dotsc , H_{m-2})\) be a parallel frame for the orthogonal complement of \(H_{1}\) in \(\mathcal {H}(P)\).

Definition 6.1

We say that \(\gamma \) is three-dimensional if the following equations hold:

$$\begin{aligned} D_{t}E&= \kappa P,\\ D_{t}H_{1}&= \tau P,\\ D_{t}H_{2}&= \cdots = D_{t}H_{m-2} =0,\\ D_{t}P&= -\kappa E -\tau H_{1}. \end{aligned}$$

It is clear that \(\gamma \) is three-dimensional if and only if W(P) is parallel in \(\mathcal {H}(P)\).

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 \(\theta \) be the rotation angle of P with respect to W(P). We know from the proof of Theorem 1.8 that if \(\gamma \) is a line of curvature and P is a closed rotation, then

$$\begin{aligned} \pm \int _{0}^{\ell } \tau \, dt = \theta (\ell ) - \theta (0) = 2n\pi \quad \hbox { for some}\ n \in \mathbb {Z}. \end{aligned}$$
(2)

On the other hand, applying Lemma 4.1, we observe that the normal curvature of \(\gamma \) with respect to \(N_{S}\) is related to the curvature \(\kappa \) by the relation

$$\begin{aligned} \kappa _{n} = \kappa (\theta ) = \kappa \cos (\theta ). \end{aligned}$$

Suppose that M is convex, so that \(\kappa _{n} > 0\). Since \(\kappa >0\), we have \(\cos (\theta ) > 0\), from which we conclude that

$$\begin{aligned} \theta (\ell ) - \theta (0) \in (-\pi , \pi ). \end{aligned}$$

Together with (2), this implies \(n =0\), as desired. \(\square \)