Namespaces
Variants
Actions

Difference between revisions of "Frame"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A set of linearly independent vectors taken in a definite order and placed at a common origin. Any three non-parallel vectors not lying in one plane can serve as a frame for the vectors in space. If the vectors building the frame are mutually orthogonal, then the frame is called orthogonal; if in this case the length of the vectors is equal to one, the frame is called orthonormal.
+
<!--
 +
f0413201.png
 +
$#A+1 = 26 n = 0
 +
$#C+1 = 26 : ~/encyclopedia/old_files/data/F041/F.0401320 Frame
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
A set of linearly independent vectors taken in a definite order and placed at a common origin. Any three non-parallel vectors not lying in one plane can serve as a frame for the vectors in space. If the vectors building the frame are mutually orthogonal, then the frame is called orthogonal; if in this case the length of the vectors is equal to one, the frame is called orthonormal.
  
 
====Comments====
 
====Comments====
 
Usually a frame is called a [[Basis|basis]] (of vectors in space). In this sense, the word  "frame"  is also used in physics (frame of reference, cf. [[Reference system|Reference system]]). For Frénet frame see [[Frénet trihedron|Frénet trihedron]].
 
Usually a frame is called a [[Basis|basis]] (of vectors in space). In this sense, the word  "frame"  is also used in physics (frame of reference, cf. [[Reference system|Reference system]]). For Frénet frame see [[Frénet trihedron|Frénet trihedron]].
  
A framing of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413201.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413202.png" /> is a vector bundle isomorphism of its tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413203.png" /> with the trivial bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413204.png" /> (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413205.png" /> is parallelizable). Using the standard basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413206.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413207.png" /> such an isomorphism defines a frame field: it assigns to every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413208.png" /> a frame, or basis, of the tangent space at that point.
+
A framing of an $  n $-
 +
dimensional differentiable manifold $  M $
 +
is a vector bundle isomorphism of its tangent bundle $  TM $
 +
with the trivial bundle $  M \times \mathbf R  ^ {n} $(
 +
so that $  M $
 +
is parallelizable). Using the standard basis $  ( e _ {1} \dots e _ {n} ) $
 +
of $  \mathbf R  ^ {n} $
 +
such an isomorphism defines a frame field: it assigns to every $  x \in M $
 +
a frame, or basis, of the tangent space at that point.
  
The frame bundle over a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f0413209.png" /> is the [[Principal fibre bundle|principal fibre bundle]] with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132010.png" /> whose fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132011.png" /> is the collection of all bases (frames) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132012.png" />, the tangent space at that point.
+
The frame bundle over a manifold $  M $
 +
is the [[Principal fibre bundle|principal fibre bundle]] with structure group $  \mathop{\rm GL} _ {n} ( \mathbf R ) $
 +
whose fibre over $  x \in M $
 +
is the collection of all bases (frames) of $  T _ {x} M $,  
 +
the tangent space at that point.
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132014.png" />-frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132016.png" /> is an ordered set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132017.png" /> linearly independent vectors. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132018.png" /> denote the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132019.png" />-frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132020.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132021.png" /> be the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132022.png" /> leaving a fixed frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132023.png" /> invariant. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132024.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132025.png" /> has a real-analytic structure. It is called the Stiefel manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132028.png" />-frames in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041320/f04132029.png" />-space.
+
A $  k $-
 +
frame $  v  ^ {k} $
 +
in $  \mathbf R  ^ {n} $
 +
is an ordered set of $  k $
 +
linearly independent vectors. Let $  V _ {n,k} $
 +
denote the set of all $  k $-
 +
frames in $  \mathbf R  ^ {n} $.  
 +
Let $  G ( k) $
 +
be the subgroup of $  \mathop{\rm GL} _ {n} ( \mathbf R ) $
 +
leaving a fixed frame $  v _ {0}  ^ {k} $
 +
invariant. Then $  V _ {n,k} = \mathop{\rm GL} _ {n} ( \mathbf R ) / G ( k) $.  
 +
Thus, $  V _ {n,k} $
 +
has a real-analytic structure. It is called the Stiefel manifold of $  k $-
 +
frames in $  n $-
 +
space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.E. Steenrod,  "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.E. Steenrod,  "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR></table>

Latest revision as of 19:39, 5 June 2020


A set of linearly independent vectors taken in a definite order and placed at a common origin. Any three non-parallel vectors not lying in one plane can serve as a frame for the vectors in space. If the vectors building the frame are mutually orthogonal, then the frame is called orthogonal; if in this case the length of the vectors is equal to one, the frame is called orthonormal.

Comments

Usually a frame is called a basis (of vectors in space). In this sense, the word "frame" is also used in physics (frame of reference, cf. Reference system). For Frénet frame see Frénet trihedron.

A framing of an $ n $- dimensional differentiable manifold $ M $ is a vector bundle isomorphism of its tangent bundle $ TM $ with the trivial bundle $ M \times \mathbf R ^ {n} $( so that $ M $ is parallelizable). Using the standard basis $ ( e _ {1} \dots e _ {n} ) $ of $ \mathbf R ^ {n} $ such an isomorphism defines a frame field: it assigns to every $ x \in M $ a frame, or basis, of the tangent space at that point.

The frame bundle over a manifold $ M $ is the principal fibre bundle with structure group $ \mathop{\rm GL} _ {n} ( \mathbf R ) $ whose fibre over $ x \in M $ is the collection of all bases (frames) of $ T _ {x} M $, the tangent space at that point.

A $ k $- frame $ v ^ {k} $ in $ \mathbf R ^ {n} $ is an ordered set of $ k $ linearly independent vectors. Let $ V _ {n,k} $ denote the set of all $ k $- frames in $ \mathbf R ^ {n} $. Let $ G ( k) $ be the subgroup of $ \mathop{\rm GL} _ {n} ( \mathbf R ) $ leaving a fixed frame $ v _ {0} ^ {k} $ invariant. Then $ V _ {n,k} = \mathop{\rm GL} _ {n} ( \mathbf R ) / G ( k) $. Thus, $ V _ {n,k} $ has a real-analytic structure. It is called the Stiefel manifold of $ k $- frames in $ n $- space.

References

[a1] N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)
How to Cite This Entry:
Frame. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frame&oldid=16777
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article