Torsion

The torsion of a curve $ \gamma $ in $ 3 $- space is a quantity characterizing the deviation of $ \gamma $ from its osculating plane. Let $ P $ be an arbitrary point on $ \gamma $ and let $ Q $ be a point near $ P $, let $ \Delta \theta $ be the angle between the osculating planes to $ \gamma $ at $ P $ and $ Q $, and let $ | \Delta s | $ be the length of the arc $ PQ $ of $ \gamma $. The absolute torsion $ | k _ {2} | $ of $ \gamma $ at $ P $ is defined as

$$ | k _ {2} | = \lim\limits _ {Q \rightarrow P } \frac{\Delta \theta }{| \Delta s | } . $$

The torsion of $ \gamma $ is defined as $ k _ {2} = \pm | k _ {2} | $, it being considered positive (negative) if an observer at $ P $ sees the osculating plane turning in the counter-clockwise (clockwise) sense as the point moves along the curve in the direction of increasing $ s $ 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 $ r = r ( s) $ is the natural parametrization of the curve, then

$$ k _ {2} = - \frac{( r ^ \prime , r ^ {\prime\prime} , r ^ {\prime\prime\prime} ) }{[ r ^ \prime , r ^ {\prime\prime} ] ^ {2} } . $$

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 $ n $- space. If the curve is sufficiently differentiable, then in this case, generically, $ n- 1 $ curvatures can be defined for it, and the last curvature can be equipped with a sign again.

The torsion of a curve in $ 3 $- 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 $ E ^ {3} $ and is defined by

$$ a = ( x _ {1} , x _ {3} , x _ {3} ^ \prime ), $$

where $ x _ {1} $ is the tangent vector to the base curve $ \Gamma $ of the strip and $ x _ {3} $ is the normal vector of the strip. The ordinary torsion $ k _ {2} $ of a curve $ \Gamma $ with non-vanishing curvature is expressed in terms of $ a $ and the normal and geodesic curvatures $ b $ and $ c $ by the following formula:

$$ k _ {2} = a + \frac{b ^ \prime c - bc ^ \prime }{b ^ {2} + c ^ {2} } . $$

The vanishing of the geodesic torsion is a characteristic property of strips of curvature, in particular for strips belonging to a surface in $ E ^ {3} $— 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 $ \nu ( M ^ {k} ) $ of a manifold $ M ^ {k} $ immersed in a Riemannian space $ V ^ {n} $. Let $ \omega _ \beta ^ \alpha $ be the connection form in $ \nu ( M ^ {k} ) $, let $ \omega _ {s} ^ \alpha $ be the Eulerian curvature forms of $ M ^ {k} $ in $ V ^ {n} $, $ s = 1 \dots k $; $ \alpha , \beta = 1 \dots n - k $. Then the forms

$$ \Omega _ {\alpha R } ^ \beta = \ d \omega _ \alpha ^ \beta - \omega _ \alpha ^ \gamma \wedge \omega _ \gamma ^ \beta $$

define the Riemannian torsion, and the forms

$$ \Omega _ {\alpha G } ^ \beta = \ \omega _ \alpha ^ {s} \wedge \omega _ {s} ^ \beta $$

the Gaussian torsion of $ M ^ {k} $ in $ V ^ {n} $. These torsions are related by the formula

$$ \Omega _ {\alpha R } ^ \beta = \ \Omega _ {\alpha G } ^ \beta + R _ {\alpha kh } ^ \beta \sigma ^ {k} \wedge \sigma ^ {h} , $$

where $ R _ {\alpha kh } ^ \beta $ are the components of the curvature tensor of $ V ^ {n} $ in the direction of a bivector tangent to $ M ^ {k} $ and $ \sigma ^ {s} $ is an orthogonal cobasis of the tangent space to $ M ^ {k} $. The tensors $ S _ {ij} ^ \beta $ obtained by decomposing the torsion forms $ \Omega _ {\alpha R } ^ \beta $( $ \Omega _ {\alpha G } ^ \beta $) in terms of the forms $ \sigma ^ {i} \wedge \sigma ^ {j} $ are known as the Gaussian and Riemannian torsions.

Example. Let $ M ^ {2} $ be a surface in the Euclidean space $ E ^ {4} $. Then the Gaussian and Riemannian torsions are equal and reduce to the single number

$$ \kappa = \frac{1}{EG - F ^ { 2 } } \left | \begin{array}{lll} E & F & G \\ L _ {1} &M _ {1} &N _ {1} \\ L _ {2} &M _ {2} &N _ {2} \\ \end{array} \right | , $$

where $ E, F, G $ are the coefficients of the first fundamental form of $ M ^ {2} $ in $ E ^ {4} $ and $ L _ {i} , M _ {i} , N _ {i} $ are the coefficients of the second fundamental form of $ M ^ {2} $ in $ E ^ {4} $. The vanishing of $ \kappa $ 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 $ \kappa = 0 $ is locally a necessary and sufficient condition for $ M ^ {2} $ to lie in a Riemannian space $ V ^ {3} $ immersed in $ E ^ {4} $, and for the normal to $ M ^ {2} $ in the tangent space to $ V ^ {3} $ to point in the direction of a principal vector of the Ricci tensor of $ V ^ {3} $. In particular, vanishing torsion is a necessary condition for $ M ^ {2} $ to be flat in $ E ^ {3} $.

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

$$ ( X, Y) \rightarrow S ( X, Y) = \nabla _ {X} Y - \nabla _ {Y} X - [ X, Y], $$

where $ X, Y $ are vector fields on $ M ^ {n} $, $ \nabla _ {X} Y $ is the covariant derivative of $ Y $ along $ X $, and $ [ X, Y] $ is the Lie bracket of $ X $ and $ Y $. Setting $ X = \partial / \partial x ^ {i} $ and $ Y = \partial / \partial x ^ {j} $ in local coordinates $ x ^ {i} $, $ i = 1 \dots n $, the transformation $ S $ is given by

$$ S \left ( \frac \partial {\partial x ^ {i} } ,\ \frac \partial {\partial x ^ {j} } \right ) = \ S _ {ij} ^ {k} \frac \partial {\partial x ^ {k} } ; $$

the tensor $ S _ {ij} ^ {k} = \Gamma _ {ij} ^ {k} - \Gamma _ {ji} ^ {k} $, where $ \Gamma _ {ji} ^ {k} $ are the components of $ \Gamma $ relative to the chosen basis, is known as the torsion tensor.

An equivalent definition of the torsion utilizes the covariant differential vector-valued $ 1 $- form $ \omega ^ {k} $ of the displacement of the connection,

$$ \Omega ^ {k} = d \omega ^ {k} + \theta _ {j} ^ {k} \wedge \omega ^ {k} , $$

which is called the torsion form; here $ \theta _ {j} ^ {k} $ are the connection forms for $ \Gamma $. In terms of the local cobasis $ dx ^ {i} $( the dual of the basis $ \partial / \partial x ^ {i} $), the form is:

$$ \Omega ^ {k} = S _ {ij} ^ {k} dx ^ {i} \wedge dx ^ {j} , $$

where $ S _ {ij} ^ {k} $ has the same meaning as before.

The torsion of an affine connection $ \Gamma $ admits the following geometrical interpretation. The evolvent of every infinitesimal contour $ L $ issuing from a point $ x \in M ^ {n} $ and returning to that point on the tangent space to $ M ^ {n} $ at $ x $ 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 $ \Omega ^ {k} $, $ k = 1 \dots n $. In other words, this vector is proportional to the bounded contour $ L $ of the two-dimensional area element with bivector $ df ^ { ij } $: $ \Omega ^ {k} = S _ {ij} ^ {k} df ^ { ij } $. 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 $ \Omega ^ {k} $ is then an analogue of the Burgers vector (see –).

Example. In a two-dimensional Riemannian space $ M ^ {2} $ with a metric connection, the torsion tensor reduces to a vector: $ S _ {ij} ^ {k} = S ^ {k} e _ {ij} $, where $ e _ {ij} $ is the metric bivector. Consider a small triangle in $ M ^ {2} $, the sides of which are geodesics of lengths $ a, b, c $, with angles $ A, B, C $. The principal part of the projection of the vector $ S ^ {k} $ at the point $ A $ on the side $ AB $ is equal to $ c - a \cos B - b \cos A $ divided by the area $ \sigma $ of the triangle, while that of the projection of the same vector on the perpendicular to $ AB $ is $ a \sin B - b \sin A $ divided by $ \sigma $. Thus, if $ M ^ {2} $ has zero torsion, the cosine and sine theorems of conventional trigonometry are valid up to quantities which are small in comparison with $ \sigma $.

The torsion of a space $ A $ is an element $ \tau ( X, A) $ of the Whitehead group $ \mathop{\rm Wh} A $ defined by the pair $ ( X, A) $, where $ A $ is a finite cellular space and the imbedding $ A \subset X $ is a homotopy equivalence. Equivalently: The torsion is an element of the Whitehead group $ \mathop{\rm Wh} \pi _ {1} $ of the fundamental group $ \pi _ {1} $. 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 $ A $ is simply connected, its torsion is zero (cf. Whitehead torsion).

If $ ( W; M _ {0} , M _ {1} ) $ is an arbitrary $ h $- cobordism, then $ \tau ( W, M _ {0} ) = \tau ( K, M _ {0} ) $, where $ K $ is the cellular space associated with a given handle decomposition of the manifold $ W $( of the manifold $ M _ {0} $), is called the torsion of the $ h $- cobordism.

Let $ M _ {f} $ be the cylinder of a cellular mapping $ f: X \rightarrow Y $ which is a homotopy equivalence (cf. Mapping cylinder). Then $ \tau ( M _ {f} , Y) = 0 $, but $ \tau ( M _ {f} , X) \in \mathop{\rm Wh} \pi _ {1} X $ does not vanish everywhere. It is defined by the formula

$$ \tau ( f ) = f _ {*} \tau ( M _ {f} , X) \in \mathop{\rm Wh} \pi _ {1} Y . $$

This element is called the torsion of the mapping $ f $( sometimes $ \tau ( M _ {f} , X) $ itself is called the torsion). If $ \tau ( f ) = 0 $, the mapping is called a simple homotopy equivalence (see ).

The torsion of a finitely-generated Abelian group $ G $ is the group $ T $ of all elements of finite order $ \nu $ in $ G $. The numbers $ \nu > 1 $ may be chosen uniquely, up to permutations, as powers of prime numbers, and they are then called the torsion coefficients of $ G $( 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 $ T( A) $ of an Abelian group, $ T( A) = \{ {a \in A } : {na = 0 \textrm{ for some } n } \} $, 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)