Namespaces
Variants
Actions

Difference between revisions of "Darboux tensor"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
d0301801.png
 +
$#A+1 = 7 n = 0
 +
$#C+1 = 7 : ~/encyclopedia/old_files/data/D030/D.0300180 Darboux tensor
 +
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 symmetric tensor of valency three,
 
A symmetric tensor of valency three,
  
<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/d030/d030180/d0301801.png" /></td> </tr></table>
+
$$
 +
\theta _ {\alpha \beta \gamma }  = b _ {\alpha \beta \gamma }  -
 +
 
 +
\frac{b _ {\alpha \beta }  K _  \gamma  + b _ {\beta \gamma }  K _  \alpha  + b _ {\gamma \alpha }  K _  \beta  }{4K}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301802.png" /> are the coefficients of the second fundamental form of the surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301803.png" /> is the Gaussian curvature, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030180/d0301805.png" /> are their covariant derivatives. G. Darboux [[#References|[1]]] was the first to investigate this tensor in special coordinates.
+
where $  b _ {\alpha \beta }  $
 +
are the coefficients of the second fundamental form of the surface, $  K $
 +
is the Gaussian curvature, and $  b _ {\alpha \beta \gamma }  $
 +
and $  K _  \alpha  $
 +
are their covariant derivatives. G. Darboux [[#References|[1]]] was the first to investigate this tensor in special coordinates.
  
 
The cubic differential form
 
The cubic differential form
  
<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/d030/d030180/d0301806.png" /></td> </tr></table>
+
$$
 +
\theta _ {\alpha \beta \gamma }  du  ^  \alpha  du  ^  \beta  du  ^  \gamma
 +
= b _ {\alpha \beta \gamma }  du  ^  \alpha  du  ^  \beta  du  ^  \gamma  +
 +
$$
  
<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/d030/d030180/d0301807.png" /></td> </tr></table>
+
$$
 +
-  
 +
\frac{3}{4}
 +
 +
\frac{K _  \gamma  }{K}
 +
b _ {\alpha
 +
\beta }  du  ^  \alpha  du  ^  \beta  du  ^  \gamma
 +
$$
  
 
is connected with the Darboux tensor. This form, evaluated for a curve on a surface, is known as the Darboux invariant. On a surface of constant negative curvature the Darboux invariant coincides with the [[Differential parameter|differential parameter]] on any one of its curves. A curve at each point of which the Darboux invariant vanishes is known as a Darboux curve. Only one real family of Darboux curves exists on a non-ruled surface of negative curvature. Three real families of Darboux curves exist on a surface of positive curvature. A surface at each point of which the Darboux tensor is defined and vanishes identically is called a Darboux surface. Darboux surfaces are second-order surfaces which are not developable on a plane.
 
is connected with the Darboux tensor. This form, evaluated for a curve on a surface, is known as the Darboux invariant. On a surface of constant negative curvature the Darboux invariant coincides with the [[Differential parameter|differential parameter]] on any one of its curves. A curve at each point of which the Darboux invariant vanishes is known as a Darboux curve. Only one real family of Darboux curves exists on a non-ruled surface of negative curvature. Three real families of Darboux curves exist on a surface of positive curvature. A surface at each point of which the Darboux tensor is defined and vanishes identically is called a Darboux surface. Darboux surfaces are second-order surfaces which are not developable on a plane.
Line 15: Line 46:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,  "Etude géométrique sur les percussions et le choc des corps"  ''Bull. Sci. Math. Ser. 2'' , '''4'''  (1880)  pp. 126–160</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1948)  pp. 210–233  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,  "Etude géométrique sur les percussions et le choc des corps"  ''Bull. Sci. Math. Ser. 2'' , '''4'''  (1880)  pp. 126–160</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,  "Foundations of the theory of surfaces in a tensor setting" , '''2''' , Moscow-Leningrad  (1948)  pp. 210–233  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Fubini,  E. Čech,  "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.P. Lane,  "A treatise on projective differential geometry" , Univ. Chicago Press  (1942)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Fubini,  E. Čech,  "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.P. Lane,  "A treatise on projective differential geometry" , Univ. Chicago Press  (1942)</TD></TR></table>

Latest revision as of 17:32, 5 June 2020


A symmetric tensor of valency three,

$$ \theta _ {\alpha \beta \gamma } = b _ {\alpha \beta \gamma } - \frac{b _ {\alpha \beta } K _ \gamma + b _ {\beta \gamma } K _ \alpha + b _ {\gamma \alpha } K _ \beta }{4K} , $$

where $ b _ {\alpha \beta } $ are the coefficients of the second fundamental form of the surface, $ K $ is the Gaussian curvature, and $ b _ {\alpha \beta \gamma } $ and $ K _ \alpha $ are their covariant derivatives. G. Darboux [1] was the first to investigate this tensor in special coordinates.

The cubic differential form

$$ \theta _ {\alpha \beta \gamma } du ^ \alpha du ^ \beta du ^ \gamma = b _ {\alpha \beta \gamma } du ^ \alpha du ^ \beta du ^ \gamma + $$

$$ - \frac{3}{4} \frac{K _ \gamma }{K} b _ {\alpha \beta } du ^ \alpha du ^ \beta du ^ \gamma $$

is connected with the Darboux tensor. This form, evaluated for a curve on a surface, is known as the Darboux invariant. On a surface of constant negative curvature the Darboux invariant coincides with the differential parameter on any one of its curves. A curve at each point of which the Darboux invariant vanishes is known as a Darboux curve. Only one real family of Darboux curves exists on a non-ruled surface of negative curvature. Three real families of Darboux curves exist on a surface of positive curvature. A surface at each point of which the Darboux tensor is defined and vanishes identically is called a Darboux surface. Darboux surfaces are second-order surfaces which are not developable on a plane.

References

[1] G. Darboux, "Etude géométrique sur les percussions et le choc des corps" Bull. Sci. Math. Ser. 2 , 4 (1880) pp. 126–160
[2] V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) pp. 210–233 (In Russian)

Comments

References

[a1] G. Fubini, E. Čech, "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars (1931)
[a2] G. Bol, "Projective Differentialgeometrie" , Vandenhoeck & Ruprecht (1954)
[a3] E.P. Lane, "A treatise on projective differential geometry" , Univ. Chicago Press (1942)
How to Cite This Entry:
Darboux tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_tensor&oldid=46582
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article