Torsion

The torsion of a curve in -space is a quantity characterizing the deviation of from its osculating plane. Let be an arbitrary point on and let be a point near , let be the angle between the osculating planes to at and , and let be the length of the arc of . The absolute torsion of at is defined as

The torsion of is defined as , it being considered positive (negative) if an observer at sees the osculating plane turning in the counter-clockwise (clockwise) sense as the point moves along the curve in the direction of increasing from the binormal vector to the principal normal vector.

For a regular (thrice continuously differentiable) curve the torsion is defined at any point where its curvature does not vanish. If is the natural parametrization of the curve, then

The torsion is sometimes called the second curvature.

The torsion and the curvature, as functions of the arc length, determine the curve up to its position in space.

A curve with vanishing torsion at each point is a planar curve.

Comments

The term "second curvature" is commonly used in higher-dimensional Frénet theory, where the curve is considered in Euclidean -space. If the curve is sufficiently differentiable, then in this case, generically, curvatures can be defined for it, and the last curvature can be equipped with a sign again.

The torsion of a curve in -space is connected with the angle of rotation of a parallel normal vector field along the curve. For a closed curve with positive curvature the angle of rotation of a parallel normal vector field along one period of the curve is given by its total torsion. This is also called the total twist of the curve.

The geodesic torsion is a generalization of the torsion of a curve; it is an invariant of a strip in the space and is defined by

where is the tangent vector to the base curve of the strip and is the normal vector of the strip. The ordinary torsion of a curve with non-vanishing curvature is expressed in terms of and the normal and geodesic curvatures and by the following formula:

The vanishing of the geodesic torsion is a characteristic property of strips of curvature, in particular for strips belonging to a surface in — see Curvature line.

Analogous concepts can be defined for strips in a Riemannian space (see , ).

The torsion of a submanifold is a generalization of the torsion of a curve, namely the curvature of the connection (cf. Connection; Connections on a manifold) induced in the normal bundle of a manifold immersed in a Riemannian space . Let be the connection form in , let be the Eulerian curvature forms of in , ; . Then the forms

define the Riemannian torsion, and the forms

the Gaussian torsion of in . These torsions are related by the formula

where are the components of the curvature tensor of in the direction of a bivector tangent to and is an orthogonal cobasis of the tangent space to . The tensors obtained by decomposing the torsion forms () in terms of the forms are known as the Gaussian and Riemannian torsions.

Example. Let be a surface in the Euclidean space . Then the Gaussian and Riemannian torsions are equal and reduce to the single number

where are the coefficients of the first fundamental form of in and are the coefficients of the second fundamental form of in . The vanishing of in some neighbourhood may be interpreted geometrically as the degeneration of the curvature ellipsoid to an interval on a straight line; there then exist two families of orthogonal curvature lines, the tangents to which correspond to the end-points of this interval. The equality is locally a necessary and sufficient condition for to lie in a Riemannian space immersed in , and for the normal to in the tangent space to to point in the direction of a principal vector of the Ricci tensor of . In particular, vanishing torsion is a necessary condition for to be flat in .

The torsion of an affine connection is a quantity characterizing the degree to which the covariant derivatives (cf. Covariant derivative) of some function on a manifold with this connection deviate from commutativity. It is defined by the transformation

where are vector fields on , is the covariant derivative of along , and is the Lie bracket of and . Setting and in local coordinates , , the transformation is given by

the tensor , where are the components of relative to the chosen basis, is known as the torsion tensor.

An equivalent definition of the torsion utilizes the covariant differential vector-valued -form of the displacement of the connection,

which is called the torsion form; here are the connection forms for . In terms of the local cobasis (the dual of the basis ), the form is:

where has the same meaning as before.

The torsion of an affine connection admits the following geometrical interpretation. The evolvent of every infinitesimal contour issuing from a point and returning to that point on the tangent space to at is no longer a closed curve. The vector difference between the end-points of the evolvent, evaluated up to second-order terms, has the components , . In other words, this vector is proportional to the bounded contour of the two-dimensional area element with bivector : . These ideas form the basis for the interpretation of an elastic medium with continuously distributed sources of internal stress in the form of displacements; the vector is then an analogue of the Burgers vector (see –).

Example. In a two-dimensional Riemannian space with a metric connection, the torsion tensor reduces to a vector: , where is the metric bivector. Consider a small triangle in , the sides of which are geodesics of lengths , with angles . The principal part of the projection of the vector at the point on the side is equal to divided by the area of the triangle, while that of the projection of the same vector on the perpendicular to is divided by . Thus, if has zero torsion, the cosine and sine theorems of conventional trigonometry are valid up to quantities which are small in comparison with .

The torsion of a space is an element of the Whitehead group defined by the pair , where is a finite cellular space and the imbedding is a homotopy equivalence. Equivalently: The torsion is an element of the Whitehead group of the fundamental group . The torsion is invariant under cellular expansions and contractions and under cellular refinements. It has been proved that the torsion is a topological invariant. If is simply connected, its torsion is zero (cf. Whitehead torsion).

If is an arbitrary -cobordism, then , where is the cellular space associated with a given handle decomposition of the manifold (of the manifold ), is called the torsion of the -cobordism.

Let be the cylinder of a cellular mapping which is a homotopy equivalence (cf. Mapping cylinder). Then , but does not vanish everywhere. It is defined by the formula

This element is called the torsion of the mapping (sometimes itself is called the torsion). If , the mapping is called a simple homotopy equivalence (see ).

The torsion of a finitely-generated Abelian group is the group of all elements of finite order in . The numbers may be chosen uniquely, up to permutations, as powers of prime numbers, and they are then called the torsion coefficients of (see [9]).

References

[1]	E. Cartan, "Leçons sur la géométrie des espaces de Riemann" , Gauthier-Villars (1946)
[2]	W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Elementare Differentialgeometrie" , 1 , Springer (1921)
[3]	Itogi Nauk. Algebra. Topol. Geom. 1969 (1971) pp. 123–168
[4]	R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Birkhäuser (1972)
[5]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)
[6a]	E. Cartan, "Sur les variétés à connexion affine et la théorie de la rélativité généralisée" Ann. Sci. Ecole Norm. Sup. (3) , 40 (1923) pp. 325–412
[6b]	E. Cartan, "Sur les variétés à connexion affine et la théorie de la rélativité généralisée" Ann. Sci. Ecole Norm. Sup. (3) , 41 (1924) pp. 1–25
[6c]	E. Cartan, "Sur les variétés à connexion affine et la théorie de la rélativité généralisée" Ann. Sci. Ecole Norm. Sup. (3) , 42 (1925) pp. 17–88
[6d]	E. Cartan, "Sur les variétés à connexion projective" Bull. Soc. Math. France , 52 (1924) pp. 205–241
[6e]	E. Cartan, "Sur les variétés à connexion conforme" Ann. Soc. Polon. Math. , 2 (1924) pp. 171–221
[7]	J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951)
[8]	C.P. Rourke, B.J. Sanderson, "Introduction to piecewise-linear topology" , Springer (1972)
[9]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)

M.I. Voitsekhovskii

Comments

The torsion subgroup of an Abelian group, , defines a functor of the category of Abelian groups into itself. For torsion in the case of a left module over an associative ring cf. Torsion submodule.

References

[a1]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a3]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145
[a4]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1–4 , Chelsea, reprint (1972)
[a5]	H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963) pp. 25; 60
[a6]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969)
[a7]	B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
[a8]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
[a9]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a10]	E. Cartan, "Oeuvres complètes" , Gauthier-Villars (1952)