sborník

25. KONFERENCE O GEOMETRII A PO C ITA COV E GRAFICE
Michal Benes
ANALYSIS OF SURFACES AT SMALL
DEFORMATIONS
Abstract
The mathematical approach to the problem of innitesimal
deformations can be presented as a part of the global dierential
geometry. A necessary and sucient condition for the
existence of the second order innitesimal bendings is determined.
Keywords
Innitesimal deformations eld, elliptic paraboloid,hyperbolic
paraboloid, Hacar's surface
1 A Mathematical Denition of Innitesimal
Deformations
Let a regular surface S be given by the vector equation
S = fR; R = r(1; 2); (1; 2) 2 g ; (1)
where is an open subset of R 2 , S : ! R 3 .
A bending of the surface S can be described as a process in where
all the single points of the surface are displaced as if they were rigid
bodies.
An innitesimal deformation (bending) of the surface S is given
by a vector eld z along r dened on tangent to a bending er at
t = 0
z(1; 2) = @er
@t ( 1; 2; t) j t=0 :
The surface S is included in the family of surfaces S t (S = S0),
expressed by the equation
S t : er(1; 2; t) = r(1; 2) + tz(1; 2) (2)
in a neighborhood of (1; 2) is an immersion for suciently small t
and
des t
2 = ds
2 + t
2 (dz:dz):
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