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''infinitesimally-small deformation''
 
''infinitesimally-small deformation''
  
A concept which first appeared in the description of the deformation of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509701.png" /> in three-dimensional Euclidean space, in which the variation of the lengths of curves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509702.png" /> is of a lower order of magnitude than the change in the spatial distance between the points of these curves. In fact, the theory of infinitesimal deformations deals with vector fields and quantities associated with them, defined at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509703.png" /> and satisfying equations which represent the linearizations of the deformation equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509704.png" />.
+
A concept which first appeared in the description of the deformation of a surface $  F $
 +
in three-dimensional Euclidean space, in which the variation of the lengths of curves on $  F $
 +
is of a lower order of magnitude than the change in the spatial distance between the points of these curves. In fact, the theory of infinitesimal deformations deals with vector fields and quantities associated with them, defined at the points of $  F $
 +
and satisfying equations which represent the linearizations of the deformation equations of $  F $.
  
Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509705.png" /> is the position vector of a deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509706.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509707.png" />, an infinitesimal deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i0509708.png" /> is characterized by the (initial) deformation rate, i.e. by the vector field
+
Thus, if $  x( u, v, t) $
 +
is the position vector of a deformation $  F _ {t} $
 +
of the surface $  F = F _ {0} $,  
 +
an infinitesimal deformation of $  F $
 +
is characterized by the (initial) deformation rate, i.e. by the vector field
  
<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/i/i050/i050970/i0509709.png" /></td> </tr></table>
+
$$
 +
z ( u, v)  = \
 +
\left .
 +
\frac{\partial  x }{\partial  t }
 +
\right | _ {t = 0 }  ,
 +
$$
  
 
which satisfies the equation
 
which satisfies the equation
  
<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/i/i050/i050970/i05097010.png" /></td> </tr></table>
+
$$
 +
( dx  dz)  = 0
 +
$$
  
 
or
 
or
  
<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/i/i050/i050970/i05097011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
( x _ {u} z _ {u} )  = \
 +
( x _ {v} z _ {v} )  = \
 +
( x _ {u} z _ {v} ) + ( x _ {v} z _ {u} )  = 0,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097012.png" /> is the position vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097013.png" />. The vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097014.png" /> is also known as the velocity field of the infinitesimal deformation or as the bending field. A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097015.png" /> can be uniquely defined such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097016.png" />. The set of points of the space described by the position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097017.png" /> is called the rotation diagram of the infinitesimal deformation. See also [[Darboux surfaces|Darboux surfaces]].
+
where $  x = x( u, v, 0) $
 +
is the position vector of $  F $.  
 +
The vector field $  z $
 +
is also known as the velocity field of the infinitesimal deformation or as the bending field. A vector $  y $
 +
can be uniquely defined such that $  dz = [ y  dx] $.  
 +
The set of points of the space described by the position vector $  y $
 +
is called the rotation diagram of the infinitesimal deformation. See also [[Darboux surfaces|Darboux surfaces]].
  
In a more general situation, the infinitesimal deformation of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097018.png" /> imbedded in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097019.png" /> represents an isometric variation of the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097020.png" />, i.e. such a vector field along the imbedding
+
In a more general situation, the infinitesimal deformation of a manifold $  M  ^ {k} $
 +
imbedded in a Riemannian space $  V  ^ {n} $
 +
represents an isometric variation of the imbedding $  i: M  ^ {k} \rightarrow V  ^ {n} $,  
 +
i.e. such a vector field along the imbedding
  
<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/i/i050/i050970/i05097021.png" /></td> </tr></table>
+
$$
 +
Z  \in  \tau ( V  ^ {n} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097022.png" /> is the tangent bundle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097023.png" />, which satisfies the equation
+
where $  \tau ( V  ^ {n} ) $
 +
is the tangent bundle to $  V  ^ {n} $,  
 +
which satisfies the equation
  
<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/i/i050/i050970/i05097024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
+
$$ \tag{1'}
 +
g ( \nabla _ {X} Z, Y) + g ( X, \nabla _ {Y} Z)  = 0
 +
$$
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097025.png" />; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097026.png" /> are vector fields tangent to the imbedding, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097027.png" /> is the Riemannian metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097029.png" /> is the covariant derivative with respect to the [[Levi-Civita connection|Levi-Civita connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097030.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097031.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097032.png" /> uniquely determines the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097033.png" /> of anti-symmetric tensors of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097034.png" /> along the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097035.png" />, satisfying the equation
+
on $  M  ^ {k} $;  
 +
here, $  X, Y \in i ^ {*} \tau ( M  ^ {k} ) $
 +
are vector fields tangent to the imbedding, $  g ( \cdot , \cdot ) $
 +
is the Riemannian metric of $  V  ^ {n} $
 +
and $  \nabla _ {X} $
 +
is the covariant derivative with respect to the [[Levi-Civita connection|Levi-Civita connection]] on $  V  ^ {n} $
 +
corresponding to $  g $.  
 +
The field $  Z $
 +
uniquely determines the field $  K _ {Z} $
 +
of anti-symmetric tensors of type $  ( 1, 1) $
 +
along the imbedding $  K _ {Z} X = \nabla _ {X} Z $,  
 +
satisfying the equation
  
<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/i/i050/i050970/i05097036.png" /></td> </tr></table>
+
$$
 +
\nabla _ {X} K _ {Z} Y -
 +
\nabla _ {Y} K _ {Z} X +
 +
K _ {Z} [ X, Y]  = \
 +
R ( X, Y) Z,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097037.png" /> is the Riemannian curvature operator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097038.png" />.
+
where $  R $
 +
is the Riemannian curvature operator of $  V  ^ {n} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097039.png" /> is induced by a [[Killing vector|Killing vector]] field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097040.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097041.png" />, then the corresponding infinitesimal deformations (and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097042.png" /> itself) are called trivial. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097043.png" /> allows only trivial infinitesimal deformations, then it is called rigid. (Cf. [[Rigidity|Rigidity]].)
+
If $  Z $
 +
is induced by a [[Killing vector|Killing vector]] field $  \xi \in \tau ( V  ^ {n} ) $,  
 +
i.e. $  Z = \xi \cdot i $,  
 +
then the corresponding infinitesimal deformations (and also $  Z $
 +
itself) are called trivial. If $  M  ^ {k} $
 +
allows only trivial infinitesimal deformations, then it is called rigid. (Cf. [[Rigidity|Rigidity]].)
  
Under a [[Geodesic mapping|geodesic mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097044.png" />, an infinitesimal deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097045.png" /> with a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097046.png" /> uniquely corresponds to an infinitesimal deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097047.png" /> with the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097048.png" />, and
+
Under a [[Geodesic mapping|geodesic mapping]] $  F: V  ^ {n} \rightarrow {\widetilde{V}  } {}  ^ {n} $,  
 +
an infinitesimal deformation of $  M  ^ {k} \subset  V  ^ {n} $
 +
with a vector field $  Z $
 +
uniquely corresponds to an infinitesimal deformation of $  F( M  ^ {k} ) \subset  {\widetilde{V}  } {}  ^ {n} $
 +
with the vector field $  \widetilde{Z}  = F ^ { * } $,  
 +
and
  
<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/i/i050/i050970/i05097049.png" /></td> </tr></table>
+
$$
 +
\widetilde{g}  ( \widetilde{Z}  , F ^ { * } ( Y))  = \psi g ( Z, Y),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097050.png" /> is the potential of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050970/i05097051.png" />. In particular, this correspondence exists under a projective transformation of the Euclidean space (the Darboux–Sauer theorem) and under a geodesic mapping of the Euclidean space into a space of constant curvature (a Pogorelov transformation).
+
where $  \psi $
 +
is the potential of the mapping $  F $.  
 +
In particular, this correspondence exists under a projective transformation of the Euclidean space (the Darboux–Sauer theorem) and under a geodesic mapping of the Euclidean space into a space of constant curvature (a Pogorelov transformation).
  
 
To isometric variations of higher orders correspond infinitesimal deformations of higher orders; unlike for the first-order infinitesimal deformations discussed above, only isolated results, mainly concerning surfaces of rotation, are available.
 
To isometric variations of higher orders correspond infinitesimal deformations of higher orders; unlike for the first-order infinitesimal deformations discussed above, only isolated results, mainly concerning surfaces of rotation, are available.
Line 44: Line 120:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Qualitative problems in the theory of deformations of surfaces"  ''Uspekhi Mat. Nauk'' , '''3''' :  2 (24)  (1948)  pp. 47–158  (In Russian)  (Translated into German as book)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Blaschke,  "Einführung in die Differentialgeometrie" , Springer  (1950)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.N. Vekua,  "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,  "Qualitative problems in the theory of deformations of surfaces"  ''Uspekhi Mat. Nauk'' , '''3''' :  2 (24)  (1948)  pp. 47–158  (In Russian)  (Translated into German as book)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,  "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Blaschke,  "Einführung in die Differentialgeometrie" , Springer  (1950)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.N. Vekua,  "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)</TD></TR>
 
+
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.W. [N.V. Efimov] Efimow,  "Flachenverbiegung im Grossen" , Akademie Verlag  (1957)  (Translated from Russian)</TD></TR>
 
+
</table>
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.W. [N.V. Efimov] Efimow,  "Flachenverbiegung im Grossen" , Akademie Verlag  (1957)  (Translated from Russian)</TD></TR></table>
 

Latest revision as of 16:00, 12 April 2023


infinitesimally-small deformation

A concept which first appeared in the description of the deformation of a surface $ F $ in three-dimensional Euclidean space, in which the variation of the lengths of curves on $ F $ is of a lower order of magnitude than the change in the spatial distance between the points of these curves. In fact, the theory of infinitesimal deformations deals with vector fields and quantities associated with them, defined at the points of $ F $ and satisfying equations which represent the linearizations of the deformation equations of $ F $.

Thus, if $ x( u, v, t) $ is the position vector of a deformation $ F _ {t} $ of the surface $ F = F _ {0} $, an infinitesimal deformation of $ F $ is characterized by the (initial) deformation rate, i.e. by the vector field

$$ z ( u, v) = \ \left . \frac{\partial x }{\partial t } \right | _ {t = 0 } , $$

which satisfies the equation

$$ ( dx dz) = 0 $$

or

$$ \tag{1 } ( x _ {u} z _ {u} ) = \ ( x _ {v} z _ {v} ) = \ ( x _ {u} z _ {v} ) + ( x _ {v} z _ {u} ) = 0, $$

where $ x = x( u, v, 0) $ is the position vector of $ F $. The vector field $ z $ is also known as the velocity field of the infinitesimal deformation or as the bending field. A vector $ y $ can be uniquely defined such that $ dz = [ y dx] $. The set of points of the space described by the position vector $ y $ is called the rotation diagram of the infinitesimal deformation. See also Darboux surfaces.

In a more general situation, the infinitesimal deformation of a manifold $ M ^ {k} $ imbedded in a Riemannian space $ V ^ {n} $ represents an isometric variation of the imbedding $ i: M ^ {k} \rightarrow V ^ {n} $, i.e. such a vector field along the imbedding

$$ Z \in \tau ( V ^ {n} ), $$

where $ \tau ( V ^ {n} ) $ is the tangent bundle to $ V ^ {n} $, which satisfies the equation

$$ \tag{1'} g ( \nabla _ {X} Z, Y) + g ( X, \nabla _ {Y} Z) = 0 $$

on $ M ^ {k} $; here, $ X, Y \in i ^ {*} \tau ( M ^ {k} ) $ are vector fields tangent to the imbedding, $ g ( \cdot , \cdot ) $ is the Riemannian metric of $ V ^ {n} $ and $ \nabla _ {X} $ is the covariant derivative with respect to the Levi-Civita connection on $ V ^ {n} $ corresponding to $ g $. The field $ Z $ uniquely determines the field $ K _ {Z} $ of anti-symmetric tensors of type $ ( 1, 1) $ along the imbedding $ K _ {Z} X = \nabla _ {X} Z $, satisfying the equation

$$ \nabla _ {X} K _ {Z} Y - \nabla _ {Y} K _ {Z} X + K _ {Z} [ X, Y] = \ R ( X, Y) Z, $$

where $ R $ is the Riemannian curvature operator of $ V ^ {n} $.

If $ Z $ is induced by a Killing vector field $ \xi \in \tau ( V ^ {n} ) $, i.e. $ Z = \xi \cdot i $, then the corresponding infinitesimal deformations (and also $ Z $ itself) are called trivial. If $ M ^ {k} $ allows only trivial infinitesimal deformations, then it is called rigid. (Cf. Rigidity.)

Under a geodesic mapping $ F: V ^ {n} \rightarrow {\widetilde{V} } {} ^ {n} $, an infinitesimal deformation of $ M ^ {k} \subset V ^ {n} $ with a vector field $ Z $ uniquely corresponds to an infinitesimal deformation of $ F( M ^ {k} ) \subset {\widetilde{V} } {} ^ {n} $ with the vector field $ \widetilde{Z} = F ^ { * } $, and

$$ \widetilde{g} ( \widetilde{Z} , F ^ { * } ( Y)) = \psi g ( Z, Y), $$

where $ \psi $ is the potential of the mapping $ F $. In particular, this correspondence exists under a projective transformation of the Euclidean space (the Darboux–Sauer theorem) and under a geodesic mapping of the Euclidean space into a space of constant curvature (a Pogorelov transformation).

To isometric variations of higher orders correspond infinitesimal deformations of higher orders; unlike for the first-order infinitesimal deformations discussed above, only isolated results, mainly concerning surfaces of rotation, are available.

The theory of infinitesimal deformations has numerous applications in mathematics and mechanics. Principal applications include problems of isometric imbedding by the method of extension along a parameter, studies of isometric surfaces in spaces of constant curvature (cf. Cohn-Vossen transformation), in problems of rigidity of shells, etc.

References

[1] N.V. Efimov, "Qualitative problems in the theory of deformations of surfaces" Uspekhi Mat. Nauk , 3 : 2 (24) (1948) pp. 47–158 (In Russian) (Translated into German as book)
[2] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian)
[3] W. Blaschke, "Einführung in die Differentialgeometrie" , Springer (1950)
[4] I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian)
[a1] N.W. [N.V. Efimov] Efimow, "Flachenverbiegung im Grossen" , Akademie Verlag (1957) (Translated from Russian)
How to Cite This Entry:
Infinitesimal deformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitesimal_deformation&oldid=19319
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article