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The torsion of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933001.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933002.png" />-space is a quantity characterizing the deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933003.png" /> from its [[Osculating plane|osculating plane]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933004.png" /> be an arbitrary point on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933005.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933006.png" /> be a point near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933007.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933008.png" /> be the angle between the osculating planes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t0933009.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330011.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330012.png" /> be the length of the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330014.png" />. The absolute torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330016.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330017.png" /> is defined as
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330018.png" /></td> </tr></table>
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The torsion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330019.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330020.png" />, it being considered positive (negative) if an observer at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330021.png" /> sees the osculating plane turning in the counter-clockwise (clockwise) sense as the point moves along the curve in the direction of increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330022.png" /> from the [[Binormal|binormal]] vector to the [[Principal normal|principal normal]] vector.
+
The torsion of a curve  $  \gamma $
 +
in $  3 $-
 +
space is a quantity characterizing the deviation of $  \gamma $
 +
from its [[Osculating plane|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
  
For a regular (thrice continuously differentiable) curve the torsion is defined at any point where its [[Curvature|curvature]] does not vanish. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330023.png" /> is the natural parametrization of the curve, then
+
$$
 +
| k _ {2} |  = \lim\limits _ {Q \rightarrow P } 
 +
\frac{\Delta \theta }{| \Delta s | }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330024.png" /></td> </tr></table>
+
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|binormal]] vector to the [[Principal normal|principal normal]] vector.
 +
 
 +
For a regular (thrice continuously differentiable) curve the torsion is defined at any point where its [[Curvature|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 is sometimes called the second curvature.
Line 14: Line 56:
  
 
A curve with vanishing torsion at each point is a planar curve.
 
A curve with vanishing torsion at each point is a planar curve.
 
 
  
 
====Comments====
 
====Comments====
The term  "second curvature"  is commonly used in higher-dimensional Frénet theory, where the curve is considered in Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330025.png" />-space. If the curve is sufficiently differentiable, then in this case, generically, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330026.png" /> curvatures can be defined for it, and the last curvature can be equipped with a sign again.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330027.png" />-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 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|strip]] in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330028.png" /> and is defined by
+
The geodesic torsion is a generalization of the torsion of a curve; it is an invariant of a [[Strip|strip]] in the space $  E  ^ {3} $
 +
and is defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330029.png" /></td> </tr></table>
+
$$
 +
= ( x _ {1} , x _ {3} , x _ {3}  ^  \prime  ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330030.png" /> is the tangent vector to the base curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330031.png" /> of the strip and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330032.png" /> is the normal vector of the strip. The ordinary torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330033.png" /> of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330034.png" /> with non-vanishing curvature is expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330035.png" /> and the normal and geodesic curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330037.png" /> by the following formula:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330038.png" /></td> </tr></table>
+
$$
 +
k _ {2}  = a +
 +
\frac{b  ^  \prime  c - bc  ^  \prime  }{b  ^ {2} + c  ^ {2} }
 +
.
 +
$$
  
The vanishing of the geodesic torsion is a characteristic property of strips of curvature, in particular for strips belonging to a surface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330039.png" /> — see [[Curvature line|Curvature line]].
+
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|Curvature line]].
  
 
Analogous concepts can be defined for strips in a Riemannian space (see , ).
 
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|Connection]]; [[Connections on a manifold|Connections on a manifold]]) induced in the normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330040.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330041.png" /> immersed in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330042.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330043.png" /> be the connection form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330044.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330045.png" /> be the Eulerian curvature forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330048.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330049.png" />. Then the forms
+
The torsion of a submanifold is a generalization of the torsion of a curve, namely the curvature of the connection (cf. [[Connection|Connection]]; [[Connections on a manifold|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330050.png" /></td> </tr></table>
+
$$
 +
\Omega _ {\alpha R }  ^  \beta  = \
 +
d \omega _  \alpha  ^  \beta  -
 +
\omega _  \alpha  ^  \gamma  \wedge
 +
\omega _  \gamma  ^  \beta
 +
$$
  
 
define the Riemannian torsion, and the forms
 
define the Riemannian torsion, and the forms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330051.png" /></td> </tr></table>
+
$$
 +
\Omega _ {\alpha G }  ^  \beta  = \
 +
\omega _  \alpha  ^ {s} \wedge
 +
\omega _ {s}  ^  \beta
 +
$$
 +
 
 +
the Gaussian torsion of  $  M  ^ {k} $
 +
in  $  V  ^ {n} $.
 +
These torsions are related by the formula
 +
 
 +
$$
 +
\Omega _ {\alpha R }  ^  \beta  = \
 +
\Omega _ {\alpha G }  ^  \beta  +
 +
R _ {\alpha kh }  ^  \beta  \sigma  ^ {k} \wedge \sigma  ^ {h} ,
 +
$$
  
the Gaussian torsion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330053.png" />. These torsions are related by the formula
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330054.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330055.png" /> are the components of the curvature tensor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330056.png" /> in the direction of a bivector tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330058.png" /> is an orthogonal cobasis of the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330059.png" />. The tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330060.png" /> obtained by decomposing the torsion forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330061.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330062.png" />) in terms of the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330063.png" /> are known as the Gaussian and Riemannian torsions.
+
$$
 +
\kappa  =
 +
\frac{1}{EG - F ^ { 2 } }
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330064.png" /> be a surface in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330065.png" />. Then the Gaussian and Riemannian torsions are equal and reduce to the single number
+
\left |
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330066.png" /></td> </tr></table>
+
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|Ricci tensor]] of  $  V  ^ {3} $.  
 +
In particular, vanishing torsion is a necessary condition for  $  M  ^ {2} $
 +
to be flat in  $  E  ^ {3} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330067.png" /> are the coefficients of the first fundamental form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330068.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330070.png" /> are the coefficients of the second fundamental form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330071.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330072.png" />. The vanishing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330073.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330074.png" /> is locally a necessary and sufficient condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330075.png" /> to lie in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330076.png" /> immersed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330077.png" />, and for the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330078.png" /> in the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330079.png" /> to point in the direction of a principal vector of the [[Ricci tensor|Ricci tensor]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330080.png" />. In particular, vanishing torsion is a necessary condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330081.png" /> to be flat in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330082.png" />.
+
The torsion of an affine connection  $  \Gamma $
 +
is a quantity characterizing the degree to which the covariant derivatives (cf. [[Covariant derivative|Covariant derivative]]) of some function on a manifold  $  M  ^ {n} $
 +
with this connection  $  \Gamma $
 +
deviate from commutativity. It is defined by the transformation
  
The torsion of an affine connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330083.png" /> is a quantity characterizing the degree to which the covariant derivatives (cf. [[Covariant derivative|Covariant derivative]]) of some function on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330084.png" /> with this connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330085.png" /> deviate from commutativity. It is defined by the transformation
+
$$
 +
( X, Y)  \rightarrow  S ( X, Y)  = \nabla _ {X} Y - \nabla _ {Y} X - [ X, Y],
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330086.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330087.png" /> are vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330089.png" /> is the covariant derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330090.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330091.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330092.png" /> is the Lie bracket of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330094.png" />. Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330096.png" /> in local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330098.png" />, the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t09330099.png" /> is given by
+
$$
 +
S \left (
 +
\frac \partial {\partial  x  ^ {i} }
 +
,\
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300100.png" /></td> </tr></table>
+
\frac \partial {\partial  x  ^ {j} }
 +
\right )  = \
 +
S _ {ij}  ^ {k}
 +
\frac \partial {\partial  x  ^ {k} }
 +
;
 +
$$
  
the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300102.png" /> are the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300103.png" /> relative to the chosen basis, is known as the torsion tensor.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300104.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300105.png" /> of the displacement of the connection,
+
An equivalent definition of the torsion utilizes the covariant differential vector-valued $  1 $-
 +
form $  \omega  ^ {k} $
 +
of the displacement of the connection,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300106.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {k}  = d \omega  ^ {k} + \theta _ {j}  ^ {k} \wedge \omega  ^ {k} ,
 +
$$
  
which is called the torsion form; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300107.png" /> are the connection forms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300108.png" />. In terms of the local cobasis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300109.png" /> (the dual of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300110.png" />), the form is:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300111.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {k}  = S _ {ij}  ^ {k}  dx  ^ {i} \wedge dx  ^ {j} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300112.png" /> has the same meaning as before.
+
where $  S _ {ij}  ^ {k} $
 +
has the same meaning as before.
  
The torsion of an affine connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300113.png" /> admits the following geometrical interpretation. The evolvent of every infinitesimal contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300114.png" /> issuing from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300115.png" /> and returning to that point on the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300116.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300117.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300119.png" />. In other words, this vector is proportional to the bounded contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300120.png" /> of the two-dimensional area element with bivector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300121.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300122.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300123.png" /> is then an analogue of the Burgers vector (see –).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300124.png" /> with a [[Metric connection|metric connection]], the torsion tensor reduces to a vector: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300125.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300126.png" /> is the metric bivector. Consider a small triangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300127.png" />, the sides of which are geodesics of lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300128.png" />, with angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300129.png" />. The principal part of the projection of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300130.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300131.png" /> on the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300132.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300133.png" /> divided by the area <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300134.png" /> of the triangle, while that of the projection of the same vector on the perpendicular to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300135.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300136.png" /> divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300137.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300138.png" /> has zero torsion, the cosine and sine theorems of conventional trigonometry are valid up to quantities which are small in comparison with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300139.png" />.
+
Example. In a two-dimensional Riemannian space $  M  ^ {2} $
 +
with a [[Metric connection|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300140.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300141.png" /> of the [[Whitehead group|Whitehead group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300142.png" /> defined by the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300143.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300144.png" /> is a finite [[Cellular space|cellular space]] and the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300145.png" /> is a homotopy equivalence. Equivalently: The torsion is an element of the Whitehead group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300146.png" /> of the fundamental group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300147.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300148.png" /> is simply connected, its torsion is zero (cf. [[Whitehead torsion|Whitehead torsion]]).
+
The torsion of a space $  A $
 +
is an element $  \tau ( X, A) $
 +
of the [[Whitehead group|Whitehead group]] $  \mathop{\rm Wh}  A $
 +
defined by the pair $  ( X, A) $,  
 +
where $  A $
 +
is a finite [[Cellular space|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|Whitehead torsion]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300149.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300151.png" />-cobordism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300152.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300153.png" /> is the cellular space associated with a given handle decomposition of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300154.png" /> (of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300155.png" />), is called the torsion of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300157.png" />-cobordism.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300158.png" /> be the cylinder of a cellular mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300159.png" /> which is a homotopy equivalence (cf. [[Mapping cylinder|Mapping cylinder]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300160.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300161.png" /> does not vanish everywhere. It is defined by the formula
+
Let $  M _ {f} $
 +
be the cylinder of a cellular mapping $  f: X \rightarrow Y $
 +
which is a homotopy equivalence (cf. [[Mapping cylinder|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300162.png" /></td> </tr></table>
+
$$
 +
\tau ( f  )  = f _ {*} \tau ( M _ {f} , X)  \in  \mathop{\rm Wh}  \pi _ {1} Y .
 +
$$
  
This element is called the torsion of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300163.png" /> (sometimes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300164.png" /> itself is called the torsion). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300165.png" />, the mapping is called a simple homotopy equivalence (see ).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300166.png" /> is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300167.png" /> of all elements of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300168.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300169.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300170.png" /> may be chosen uniquely, up to permutations, as powers of prime numbers, and they are then called the torsion coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300171.png" /> (see [[#References|[9]]]).
+
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 [[#References|[9]]]).
  
 
====References====
 
====References====
Line 96: Line 309:
  
 
====Comments====
 
====Comments====
The torsion subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300172.png" /> of an Abelian group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093300/t093300173.png" />, defines a [[Functor|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|Torsion submodule]].
+
The torsion subgroup $  T( A) $
 +
of an Abelian group, $  T( A) = \{ {a \in A } : {na = 0 \textrm{ for  some  }  n } \} $,  
 +
defines a [[Functor|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|Torsion submodule]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.W. Guggenheimer,  "Differential geometry" , McGraw-Hill  (1963)  pp. 25; 60</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1969)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B. O'Neill,  "Elementary differential geometry" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  E. Cartan,  "Oeuvres complètes" , Gauthier-Villars  (1952)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  B. Gostiaux,  "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.P. Do Carmo,  "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H.W. Guggenheimer,  "Differential geometry" , McGraw-Hill  (1963)  pp. 25; 60</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1–2''' , Interscience  (1963–1969)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B. O'Neill,  "Elementary differential geometry" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  pp. 1–5</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  E. Cartan,  "Oeuvres complètes" , Gauthier-Villars  (1952)</TD></TR></table>

Revision as of 08:26, 6 June 2020


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

$$ \kappa = \frac{1}{EG - F ^ { 2 } } \left | where $ E, F, G $ are the coefficients of the first fundamental form of $ M ^ {2} $ in $ E ^ {4} $ and $ L _ {i} , M _ {i} , N _ {i} $ are the coefficients of the second fundamental form of $ M ^ {2} $ in $ E ^ {4} $. The vanishing of $ \kappa $ in some neighbourhood may be interpreted geometrically as the degeneration of the curvature ellipsoid to an interval on a straight line; there then exist two families of orthogonal curvature lines, the tangents to which correspond to the end-points of this interval. The equality $ \kappa = 0 $ is locally a necessary and sufficient condition for $ M ^ {2} $ to lie in a Riemannian space $ V ^ {3} $ immersed in $ E ^ {4} $, and for the normal to $ M ^ {2} $ in the tangent space to $ V ^ {3} $ to point in the direction of a principal vector of the [[Ricci tensor|Ricci tensor]] of $ V ^ {3} $. In particular, vanishing torsion is a necessary condition for $ M ^ {2} $ to be flat in $ E ^ {3} $. The torsion of an affine connection $ \Gamma $ is a quantity characterizing the degree to which the covariant derivatives (cf. [[Covariant derivative|Covariant derivative]]) of some function on a manifold $ M ^ {n} $ with this connection $ \Gamma $ deviate from commutativity. It is defined by the transformation $$ ( 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|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|Whitehead group]] $ \mathop{\rm Wh} A $ defined by the pair $ ( X, A) $, where $ A $ is a finite [[Cellular space|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|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|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)
How to Cite This Entry:
Torsion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Torsion&oldid=48998
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article