# 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)
How to Cite This Entry:
Egorov system of surfaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Egorov_system_of_surfaces&oldid=46792
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article