Infinitesimal deformation

infinitesimally-small deformation

A concept which first appeared in the description of the deformation of a surface $F$ in three-dimensional Euclidean space, in which the variation of the lengths of curves on $F$ is of a lower order of magnitude than the change in the spatial distance between the points of these curves. In fact, the theory of infinitesimal deformations deals with vector fields and quantities associated with them, defined at the points of $F$ and satisfying equations which represent the linearizations of the deformation equations of $F$.

Thus, if $x( u, v, t)$ is the position vector of a deformation $F _ {t}$ of the surface $F = F _ {0}$, an infinitesimal deformation of $F$ is characterized by the (initial) deformation rate, i.e. by the vector field

$$z ( u, v) = \ \left . \frac{\partial x }{\partial t } \right | _ {t = 0 } ,$$

which satisfies the equation

$$( dx dz) = 0$$

or

$$\tag{1 } ( x _ {u} z _ {u} ) = \ ( x _ {v} z _ {v} ) = \ ( x _ {u} z _ {v} ) + ( x _ {v} z _ {u} ) = 0,$$

where $x = x( u, v, 0)$ is the position vector of $F$. The vector field $z$ is also known as the velocity field of the infinitesimal deformation or as the bending field. A vector $y$ can be uniquely defined such that $dz = [ y dx]$. The set of points of the space described by the position vector $y$ is called the rotation diagram of the infinitesimal deformation. See also Darboux surfaces.

In a more general situation, the infinitesimal deformation of a manifold $M ^ {k}$ imbedded in a Riemannian space $V ^ {n}$ represents an isometric variation of the imbedding $i: M ^ {k} \rightarrow V ^ {n}$, i.e. such a vector field along the imbedding

$$Z \in \tau ( V ^ {n} ),$$

where $\tau ( V ^ {n} )$ is the tangent bundle to $V ^ {n}$, which satisfies the equation

$$\tag{1'} g ( \nabla _ {X} Z, Y) + g ( X, \nabla _ {Y} Z) = 0$$

on $M ^ {k}$; here, $X, Y \in i ^ {*} \tau ( M ^ {k} )$ are vector fields tangent to the imbedding, $g ( \cdot , \cdot )$ is the Riemannian metric of $V ^ {n}$ and $\nabla _ {X}$ is the covariant derivative with respect to the Levi-Civita connection on $V ^ {n}$ corresponding to $g$. The field $Z$ uniquely determines the field $K _ {Z}$ of anti-symmetric tensors of type $( 1, 1)$ along the imbedding $K _ {Z} X = \nabla _ {X} Z$, satisfying the equation

$$\nabla _ {X} K _ {Z} Y - \nabla _ {Y} K _ {Z} X + K _ {Z} [ X, Y] = \ R ( X, Y) Z,$$

where $R$ is the Riemannian curvature operator of $V ^ {n}$.

If $Z$ is induced by a Killing vector field $\xi \in \tau ( V ^ {n} )$, i.e. $Z = \xi \cdot i$, then the corresponding infinitesimal deformations (and also $Z$ itself) are called trivial. If $M ^ {k}$ allows only trivial infinitesimal deformations, then it is called rigid. (Cf. Rigidity.)

Under a geodesic mapping $F: V ^ {n} \rightarrow {\widetilde{V} } {} ^ {n}$, an infinitesimal deformation of $M ^ {k} \subset V ^ {n}$ with a vector field $Z$ uniquely corresponds to an infinitesimal deformation of $F( M ^ {k} ) \subset {\widetilde{V} } {} ^ {n}$ with the vector field $\widetilde{Z} = F ^ { * }$, and

$$\widetilde{g} ( \widetilde{Z} , F ^ { * } ( Y)) = \psi g ( Z, Y),$$

where $\psi$ is the potential of the mapping $F$. In particular, this correspondence exists under a projective transformation of the Euclidean space (the Darboux–Sauer theorem) and under a geodesic mapping of the Euclidean space into a space of constant curvature (a Pogorelov transformation).

To isometric variations of higher orders correspond infinitesimal deformations of higher orders; unlike for the first-order infinitesimal deformations discussed above, only isolated results, mainly concerning surfaces of rotation, are available.

The theory of infinitesimal deformations has numerous applications in mathematics and mechanics. Principal applications include problems of isometric imbedding by the method of extension along a parameter, studies of isometric surfaces in spaces of constant curvature (cf. Cohn-Vossen transformation), in problems of rigidity of shells, etc.

References

 [1] N.V. Efimov, "Qualitative problems in the theory of deformations of surfaces" Uspekhi Mat. Nauk , 3 : 2 (24) (1948) pp. 47–158 (In Russian) (Translated into German as book) [2] A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian) [3] W. Blaschke, "Einführung in die Differentialgeometrie" , Springer (1950) [4] I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian)