Abstract
A curve \(\gamma \) 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 \(\gamma \) lies on an oriented hypersurface S of M, we say that \(\gamma \) is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that \(\gamma \) is three-dimensional and closed. We show that if \(\gamma \) is a well-positioned line of curvature of S, then its total torsion is an integer multiple of \(2\pi \); and that, conversely, if the total torsion of \(\gamma \) is an integer multiple of \(2\pi \), then there exists an oriented hypersurface of M in which \(\gamma \) is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of \(\gamma \) vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.
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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
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)
\(\langle H, \gamma ' \rangle = \langle H, N \rangle = 0\), i.e., \(H \in \Gamma (\mathcal {H})\);
-
(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
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)
If \(\dim M = 3\), then any unit normal vector field along \(\gamma \) is a parallel rotation of N.
-
(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
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
accordingly, we write
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
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
Note that, since \((E, H_{1}, \dotsc , H_{m-2}, N)\) is orthonormal, the following equations hold for all \(j =1, \dotsc , m-2\):
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
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
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
while the vector fields
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
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
it follows that, when \(N_{S}(\theta )\) is a closed rotation of \(N_{S}\),
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
where
(Note that \(N(\theta )\) is a parallel rotation of N.)
Define a map \(\sigma :[0, \ell ] \times \mathbb {R}^{m-1} \rightarrow M\) by
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
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:
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
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
Suppose that M is convex, so that \(\kappa _{n} > 0\). Since \(\kappa >0\), we have \(\cos (\theta ) > 0\), from which we conclude that
Together with (2), this implies \(n =0\), as desired. \(\square \)
References
Costa, S.R., Romero-Fuster, M.D.C.: Nowhere vanishing torsion closed curves always hide twice. Geom. Dedicata. 66(1), 1–17 (1997)
da Silva, L.C.B., da Silva, J.D.: Characterization of manifolds of constant curvature by spherical curves. Ann. Mat. Pura Appl. (4) 199(1), 217–229 (2020)
Fenchel, W.: Über einen Jacobischen Satz der Kurventheorie. Tôhoku Math. J. 39, 95–97 (1934)
Geppert, H.: Sopra una caratterizzazione della sfera. Ann. Mat. Pura Appl. (4) 20, 59–66 (1941)
Ghomi, M.: Boundary torsion and convex caps of locally convex surfaces. J. Differ. Geom. 105(3), 427–487 (2017)
Ghomi, M.: Torsion of locally convex curves. Proc. Am. Math. Soc. 147(4), 1699–1707 (2019)
Millman, R.S., Parker, G.D.: Elements of Differential Geometry. Prentice-Hall, Englewood Cliffs (1977)
Pansonato, C.C., Costa, S.I.R.: Total torsion of curves in three-dimensional manifolds. Geom. Dedicata. 136, 111–121 (2008)
Qin, Y., Li, S.: Total torsion of closed lines of curvature. Bull. Austral. Math. Soc. 65(1), 73–78 (2002)
Santaló, L.A.: Algunas propriedades de las curvas esféricas y una característica de la esfera. Rev. Mat. Hisp.-Amer. (2) 10, 9–12 (1935)
Scherrer, W.: Eine Kennzeichnung der Kugel. Vierteljschr. Naturforsch. Ges. Zürich 85, 40–46 (1940)
Sedykh, V.D.: Four vertices of a convex space curve. Bull. Lond. Math. Soc. 26(2), 177–180 (1994)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 4, 3rd edn. Publish or Perish, Houston (1999)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 3, 3rd edn. Publish or Perish, Houston (1999)
Yin, S., Zheng, D.: The curvature and torsion of curves in a surface. J. Geom. 108(3), 1085–1090 (2017)
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Raffaelli, M. Total torsion of three-dimensional lines of curvature. Geom Dedicata 217, 96 (2023). https://doi.org/10.1007/s10711-023-00833-8
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DOI: https://doi.org/10.1007/s10711-023-00833-8