Difference between revisions of "Second fundamental form"
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)</TD></TR></table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) {{MR|0350630}} {{ZBL|0264.53001}} </TD></TR></table> |
Revision as of 12:13, 27 September 2012
of a surface
The quadratic form in the differentials of the coordinates on the surface which characterizes the local structure of the surface in a neighbourhood of an ordinary point. Let the surface be given by the equation
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where and
are internal coordinates on the surface; let
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be the differential of the position vector along a chosen direction
of displacement from a point
to a point
(see Fig.). Let
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be the unit normal vector to the surface at the point (here
if the vector triplet
has right orientation, and
in the opposite case). The double principal linear part
of the deviation
of the point
on the surface from the tangent plane at the point
is
![]() |
![]() |
it is known as the second fundamental form of the surface.
Figure: s083700a
The coefficients of the second fundamental form are usually denoted by
![]() |
or, in tensor notation,
![]() |
The tensor is called the second fundamental tensor of the surface.
See Fundamental forms of a surface for the connection between the second fundamental form and other surface forms.
Comments
References
[a1] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) MR0350630 Zbl 0264.53001 |
Second fundamental form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second_fundamental_form&oldid=12634