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One of the fundamental concepts in modern differential geometry including the specific structures studied in classical differential geometry. It is defined for a given differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321601.png" /> as a differentiable section in a [[Fibre space|fibre space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321602.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321603.png" /> associated with a certain principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321604.png" /> or, according to another terminology, as a differentiable field of geometric objects on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321605.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321606.png" /> is some differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321607.png" />-space where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321608.png" /> is the structure Lie group of the principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d0321609.png" /> or, in another terminology, the representation space of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216010.png" />.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216011.png" /> is the principal bundle of frames in the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216013.png" /> is some closed subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216014.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216015.png" /> is the [[Homogeneous space|homogeneous space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216016.png" />, the corresponding differential-geometric structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216017.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216019.png" />-structure or an infinitesimal structure of the first order. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216020.png" /> consists of those linear transformations (elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216021.png" />) which leave an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216022.png" />-dimensional space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216023.png" /> invariant, the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216024.png" />-structure defines a distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216025.png" />-dimensional subspaces on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216027.png" /> is the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216028.png" /> — the subgroup of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216029.png" /> which preserve the scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216030.png" /> —, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216031.png" />-structure is a Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216032.png" />, i.e. the field of a positive-definite symmetric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216033.png" />. In a similar manner, almost-complex and complex structures are special cases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216034.png" />-structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216035.png" />. A generalization of the concept of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216036.png" />-structure is an infinitesimal structure of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216039.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216041.png" />-structure of a higher order); here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216042.png" /> is the principal bundle of frames of the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216044.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216045.png" /> is a closed subgroup of its structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216046.png" />.
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All kinds of connections (cf. [[Connection|Connection]]) are important special cases of differential-geometric structures. For instance, a connection in a principal bundle is obtained if the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216047.png" /> is played by the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216048.png" /> of some principal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216049.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216050.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216051.png" /> is the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216052.png" />-dimensional, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216053.png" />, subspaces complementary to the tangent spaces of the fibres which is invariant with respect to the action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216054.png" /> of the structure group of the bundle. Connections on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216055.png" /> are special cases of differential-geometric structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216056.png" />, but more general ones than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216057.png" />-structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216058.png" />. For instance, an [[Affine connection|affine connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216059.png" />, definable by a field of connection objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216060.png" />, is obtained as the differential-geometric structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216061.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216062.png" /> is the principal bundle of frames of second order, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216063.png" /> is its structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216064.png" />, and the representation space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216065.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216066.png" /> is the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216067.png" /> with coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216068.png" />, where the representation is defined by the formulas
+
One of the fundamental concepts in modern differential geometry including the specific structures studied in classical differential geometry. It is defined for a given differentiable manifold  $  M  ^ {n} $
 +
as a differentiable section in a [[Fibre space|fibre space]] $  ( X _ {F} , p _ {F} , M  ^ {n} ) $
 +
with base  $  M  ^ {n} $
 +
associated with a certain principal bundle $  ( X , p , M  ^ {n} ) $
 +
or, according to another terminology, as a differentiable field of geometric objects on $  M  ^ {n} $.  
 +
Here  $  F $
 +
is some differentiable  $  \mathfrak G $-
 +
space where  $  \mathfrak G $
 +
is the structure Lie group of the principal bundle $  ( X , p , M  ^ {n} ) $
 +
or, in another terminology, the representation space of the Lie group  $  \mathfrak G $.
  
<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/d/d032/d032160/d03216069.png" /></td> </tr></table>
+
If  $  ( X , p , M  ^ {n} ) $
 +
is the principal bundle of frames in the tangent space to  $  M  ^ {n} $,
 +
$  G $
 +
is some closed subgroup in  $  \mathfrak G = \mathop{\rm GL} ( n, \mathbf R ) $,
 +
and  $  F $
 +
is the [[Homogeneous space|homogeneous space]]  $  \mathfrak G / G $,
 +
the corresponding differential-geometric structure on  $  M  ^ {n} $
 +
is called a  $  G $-
 +
structure or an infinitesimal structure of the first order. For example, if  $  G $
 +
consists of those linear transformations (elements of  $  \mathop{\rm GL} ( n , \mathbf R ) $)
 +
which leave an  $  m $-
 +
dimensional space in  $  \mathbf R  ^ {n} $
 +
invariant, the corresponding  $  G $-
 +
structure defines a distribution of  $  m $-
 +
dimensional subspaces on  $  M  ^ {n} $.
 +
If  $  G $
 +
is the orthogonal group  $  O ( n , \mathbf R ) $—
 +
the subgroup of elements of  $  \mathop{\rm GL} ( n , \mathbf R ) $
 +
which preserve the scalar product in  $  \mathbf R  ^ {n} $—,
 +
then the  $  G $-
 +
structure is a Riemannian metric on  $  M  ^ {n} $,
 +
i.e. the field of a positive-definite symmetric tensor  $  g _ {ij} $.
 +
In a similar manner, almost-complex and complex structures are special cases of  $  G $-
 +
structures on  $  M  ^ {n} $.
 +
A generalization of the concept of a  $  G $-
 +
structure is an infinitesimal structure of order  $  r $,
 +
$  r > 1 $(
 +
or  $  G $-
 +
structure of a higher order); here  $  ( X , p , M  ^ {n} ) $
 +
is the principal bundle of frames of the order  $  r $
 +
on  $  M  ^ {n} $,
 +
and  $  G $
 +
is a closed subgroup of its structure group  $  D _ {n}  ^ {r} $.
 +
 
 +
All kinds of connections (cf. [[Connection|Connection]]) are important special cases of differential-geometric structures. For instance, a connection in a principal bundle is obtained if the role of  $  M  ^ {n} $
 +
is played by the space  $  P $
 +
of some principal bundle  $  ( P , p , B ) $,
 +
and the  $  G $-
 +
structure on  $  P $
 +
is the distribution of  $  m $-
 +
dimensional,  $  m = \mathop{\rm dim}  P -  \mathop{\rm dim}  B $,
 +
subspaces complementary to the tangent spaces of the fibres which is invariant with respect to the action on  $  P $
 +
of the structure group of the bundle. Connections on a manifold  $  M  ^ {n} $
 +
are special cases of differential-geometric structures on  $  M  ^ {n} $,
 +
but more general ones than  $  G $-
 +
structures on  $  M  ^ {n} $.  
 +
For instance, an [[Affine connection|affine connection]] on  $  M  ^ {n} $,
 +
definable by a field of connection objects  $  \Gamma _ {ij}  ^ {k} ( x) $,
 +
is obtained as the differential-geometric structure on  $  M  ^ {n} $
 +
for which  $  ( X , p , M  ^ {n} ) $
 +
is the principal bundle of frames of second order,  $  \mathfrak G $
 +
is its structure group  $  D _ {n}  ^ {2} $,
 +
and the representation space  $  F $
 +
of  $  D _ {n}  ^ {2} $
 +
is the space  $  \mathbf R  ^ {3n} $
 +
with coordinates  $  \Gamma _ {ij}  ^ {k} $,
 +
where the representation is defined by the formulas
 +
 
 +
$$
 +
\overline \Gamma \; {} _ {st}  ^ {r}  =  ( A _ {s}  ^ {i}
 +
A _ {t}  ^ {j} \Gamma _ {ij}  ^ {k} +
 +
A _ {st}  ^ {k} ) \overline{A}\; {} _ {k}  ^ {r} ,
 +
$$
  
 
where
 
where
  
<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/d/d032/d032160/d03216070.png" /></td> </tr></table>
+
$$
 +
A _ {k}  ^ {r}  = \left (
 +
\frac{\partial  x  ^ {r} }{\partial  \overline{x}\; {}  ^ {k} }
 +
\right ) _ {0} ,\  A _ {st}  ^ {k}  = \
 +
\left (
 +
\frac{\partial  ^ {2} x  ^ {k} }{\partial  \overline{x}\; {}  ^ {s} \partial  x
 +
bar {}  ^ {t} }
 +
\right ) _ {0}  $$
  
are the coordinates of an element of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216071.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216072.png" />. In the case of a projective connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216073.png" /> one deals with a certain representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216074.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216075.png" />, while in cases of connections of a higher order, one deals with representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032160/d03216076.png" />. By this approach the theory of differential-geometric structures becomes closely related to the theory of geometric objects (Cf. [[Geometric objects, theory of|Geometric objects, theory of]]).
+
are the coordinates of an element of the group $  D _ {n}  ^ {2} $,  
 +
and $  A _ {k}  ^ {r} \overline{A}\; {} _ {t}  ^ {k} = \delta _ {t}  ^ {r} $.  
 +
In the case of a projective connection on $  M  ^ {n} $
 +
one deals with a certain representation of $  D _ {n}  ^ {3} $
 +
in $  \mathbf R ^ {3 ( n+ 1 ) } $,  
 +
while in cases of connections of a higher order, one deals with representations of $  D _ {n}  ^ {r} $.  
 +
By this approach the theory of differential-geometric structures becomes closely related to the theory of geometric objects (Cf. [[Geometric objects, theory of|Geometric objects, theory of]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Veblen,  J.H.C. Whitehead,  "The foundations of differential geometry" , Cambridge Univ. Press  (1932)  (Appendix by V.V. Vagner in the Russian translation)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Laptev,  "Differential geometry of imbedded manifolds. Group-theoretical method of differential geometric investigation"  ''Trudy Moskov. Mat. Obshch.'' , '''2'''  (1953)  pp. 275–382  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Veblen,  J.H.C. Whitehead,  "The foundations of differential geometry" , Cambridge Univ. Press  (1932)  (Appendix by V.V. Vagner in the Russian translation)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.F. Laptev,  "Differential geometry of imbedded manifolds. Group-theoretical method of differential geometric investigation"  ''Trudy Moskov. Mat. Obshch.'' , '''2'''  (1953)  pp. 275–382  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1972–1975)  pp. 1–5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1972–1975)  pp. 1–5</TD></TR></table>

Latest revision as of 17:33, 5 June 2020


One of the fundamental concepts in modern differential geometry including the specific structures studied in classical differential geometry. It is defined for a given differentiable manifold $ M ^ {n} $ as a differentiable section in a fibre space $ ( X _ {F} , p _ {F} , M ^ {n} ) $ with base $ M ^ {n} $ associated with a certain principal bundle $ ( X , p , M ^ {n} ) $ or, according to another terminology, as a differentiable field of geometric objects on $ M ^ {n} $. Here $ F $ is some differentiable $ \mathfrak G $- space where $ \mathfrak G $ is the structure Lie group of the principal bundle $ ( X , p , M ^ {n} ) $ or, in another terminology, the representation space of the Lie group $ \mathfrak G $.

If $ ( X , p , M ^ {n} ) $ is the principal bundle of frames in the tangent space to $ M ^ {n} $, $ G $ is some closed subgroup in $ \mathfrak G = \mathop{\rm GL} ( n, \mathbf R ) $, and $ F $ is the homogeneous space $ \mathfrak G / G $, the corresponding differential-geometric structure on $ M ^ {n} $ is called a $ G $- structure or an infinitesimal structure of the first order. For example, if $ G $ consists of those linear transformations (elements of $ \mathop{\rm GL} ( n , \mathbf R ) $) which leave an $ m $- dimensional space in $ \mathbf R ^ {n} $ invariant, the corresponding $ G $- structure defines a distribution of $ m $- dimensional subspaces on $ M ^ {n} $. If $ G $ is the orthogonal group $ O ( n , \mathbf R ) $— the subgroup of elements of $ \mathop{\rm GL} ( n , \mathbf R ) $ which preserve the scalar product in $ \mathbf R ^ {n} $—, then the $ G $- structure is a Riemannian metric on $ M ^ {n} $, i.e. the field of a positive-definite symmetric tensor $ g _ {ij} $. In a similar manner, almost-complex and complex structures are special cases of $ G $- structures on $ M ^ {n} $. A generalization of the concept of a $ G $- structure is an infinitesimal structure of order $ r $, $ r > 1 $( or $ G $- structure of a higher order); here $ ( X , p , M ^ {n} ) $ is the principal bundle of frames of the order $ r $ on $ M ^ {n} $, and $ G $ is a closed subgroup of its structure group $ D _ {n} ^ {r} $.

All kinds of connections (cf. Connection) are important special cases of differential-geometric structures. For instance, a connection in a principal bundle is obtained if the role of $ M ^ {n} $ is played by the space $ P $ of some principal bundle $ ( P , p , B ) $, and the $ G $- structure on $ P $ is the distribution of $ m $- dimensional, $ m = \mathop{\rm dim} P - \mathop{\rm dim} B $, subspaces complementary to the tangent spaces of the fibres which is invariant with respect to the action on $ P $ of the structure group of the bundle. Connections on a manifold $ M ^ {n} $ are special cases of differential-geometric structures on $ M ^ {n} $, but more general ones than $ G $- structures on $ M ^ {n} $. For instance, an affine connection on $ M ^ {n} $, definable by a field of connection objects $ \Gamma _ {ij} ^ {k} ( x) $, is obtained as the differential-geometric structure on $ M ^ {n} $ for which $ ( X , p , M ^ {n} ) $ is the principal bundle of frames of second order, $ \mathfrak G $ is its structure group $ D _ {n} ^ {2} $, and the representation space $ F $ of $ D _ {n} ^ {2} $ is the space $ \mathbf R ^ {3n} $ with coordinates $ \Gamma _ {ij} ^ {k} $, where the representation is defined by the formulas

$$ \overline \Gamma \; {} _ {st} ^ {r} = ( A _ {s} ^ {i} A _ {t} ^ {j} \Gamma _ {ij} ^ {k} + A _ {st} ^ {k} ) \overline{A}\; {} _ {k} ^ {r} , $$

where

$$ A _ {k} ^ {r} = \left ( \frac{\partial x ^ {r} }{\partial \overline{x}\; {} ^ {k} } \right ) _ {0} ,\ A _ {st} ^ {k} = \ \left ( \frac{\partial ^ {2} x ^ {k} }{\partial \overline{x}\; {} ^ {s} \partial x bar {} ^ {t} } \right ) _ {0} $$

are the coordinates of an element of the group $ D _ {n} ^ {2} $, and $ A _ {k} ^ {r} \overline{A}\; {} _ {t} ^ {k} = \delta _ {t} ^ {r} $. In the case of a projective connection on $ M ^ {n} $ one deals with a certain representation of $ D _ {n} ^ {3} $ in $ \mathbf R ^ {3 ( n+ 1 ) } $, while in cases of connections of a higher order, one deals with representations of $ D _ {n} ^ {r} $. By this approach the theory of differential-geometric structures becomes closely related to the theory of geometric objects (Cf. Geometric objects, theory of).

References

[1] O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932) (Appendix by V.V. Vagner in the Russian translation)
[2] G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential geometric investigation" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian)
[3] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[4] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)

Comments

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1972–1975) pp. 1–5
How to Cite This Entry:
Differential-geometric structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential-geometric_structure&oldid=15582
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article