Egorov system of surfaces

A tri-orthogonal system $ \Sigma $ consisting of so-called potential surfaces (cf. Potential net), named after D.F. Egorov, who in 1901 (see [1]) considered their general theory in detail (under the name of potential systems) and gave numerous examples of systems of this type. An Egorov system $ \Sigma $ can be defined as a system admitting a (one-parameter) group of transformations taking $ \Sigma $ into itself in such a way that the normals at corresponding points of $ \Sigma $ remain parallel. The stationary flow of a fluid with a velocity potential and carrying the surfaces of an Egorov system provides a mechanical interpretation of this group.

Let

$$ u ^ {i} ( x , y , z) = \textrm{ const } ,\ i = 1 , 2 , 3, $$

be the equations of the surfaces forming an Egorov system $ \Sigma $; let $ H _ {i} $ be the Lamé coefficients appearing in the expression for the square of the line element of the space in the curvilinear coordinates $ \{ u ^ {i} \} $:

$$ ds ^ {2} = \sum _ { i= 1} ^ { 3 } H _ {1} ^ {2} ( du ^ {i} ) ^ {2} , $$

let $ P _ {i} $ be the distance between the origin and the three tangent planes to $ \Sigma $, let $ R _ {ik} $ be the principal radii of curvature of the surfaces $ u ^ {i} = \textrm{ const } $, corresponding to the principal direction $ H _ {k} du ^ {k} $, and let $ \beta _ {ik} = - H _ {k} / R _ {ik} $ be the quantities appearing in the expression for the line elements $ d \sigma _ {i} $ of the spherical images (cf. Spherical map) of the surfaces:

$$ ( d \sigma _ {i} ) ^ {2} = \beta _ {ik} ^ {2} ( du ^ {k} ) ^ {2} + \beta _ {il} ^ {2} ( du ^ {l} ) ^ {2} ,\ i \neq k \neq l . $$

The functions $ P _ {i} $ and $ H _ {i} $ satisfy the same system of equations:

$$ \frac{\partial \theta _ {i} }{\partial u ^ {k} } = \beta _ {ik} \theta _ {k} . $$

The solutions of these equations define two other Egorov systems, $ \Sigma _ {1} $ and $ \Sigma _ {- 1} $, with the same spherical images, for which

$$ P _ {i} ^ {( 1)} = H _ {i} ,\ H _ {i} ^ {(- 1)} = P _ {i} . $$

Continuing this transformation in both directions gives a series of Egorov systems (the Egorov series)

$$ {} \dots, \Sigma _ {- 2 }, \Sigma _ {- 1} , \Sigma , \Sigma _ {1} , \Sigma _ {2} ,\dots $$

with the same spherical image, in which each $ \Sigma _ {k+ 1} $ is obtained from the previous $ \Sigma _ {k} $ by the formula

$$ P _ {i} ^ {( k+ 1)} = H _ {i} ^ {( k)} . $$

In general, the search for the spherical image of an Egorov system $ \Sigma $ reduces to the investigation of a potential system on the sphere: Any such system may be taken as the spherical image of one of the three families forming $ \Sigma $.

An Egorov system $ \Sigma $ is characterized by the fact that

$$ H _ {i} ^ {2} = \frac{\partial \omega }{\partial u ^ {i} } , $$

where $ \omega $ is a function having the meaning of velocity potential for the corresponding flow, that is, $ u ^ {i} = \textrm{ const } $ are the potential surfaces. Thus, for any potential surface $ S $, there is an Egorov system $ \Sigma $ containing $ S $. The tangent to the line of intersection of any surface $ \omega = \textrm{ const } $ with the surface $ u ^ {i} = \textrm{ const } $ at any point is parallel to the ray $ l ^ {i} $ joining the centres of geodesic curvature of the lines of curvature of the surface $ u ^ {i} = \textrm{ const } $; at each point of space the three rays $ l ^ {1} , l ^ {2} , l ^ {3} $ are parallel to a common plane — the tangent plane to the surface $ \omega = \textrm{ const } $, and the osculating planes of the coordinate lines pass through a common straight line. The quantities $ \beta _ {ik} $ and $ R _ {ik} $ for an Egorov system satisfy the relations:

$$ R _ {12} R _ {23} R _ {31} = R _ {13} R _ {32} R _ {21} ,\ \beta _ {ik} = \beta _ {ki} $$

(the symmetry of $ \beta _ {ik} $ is also a necessary and sufficient condition for a tri-orthogonal system to be an Egorov system).

References

[1]	D.F. Egorov, "Papers in differential geometry" , Moscow (1970) (In Russian)