# 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.

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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 ).

How to Cite This Entry:
Torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion&oldid=49631
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article