Abstract
A general form of the equation of a curvilinear three-web admitting a one-parameter family of automorphisms (\(AW\)-webs) is found. It is proved that the trajectories of automorphisms of an \(AW\)-web are geodesics of its Chern connection. All \(AW\)-webs are found for which one of the covariant derivatives of curvature is zero.
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1. An arbitrary curvilinear three-web does not, generally speaking, admit automorphisms. A parallel three-web admits a three-parameter group of automorphisms. Indeed, a parallel three-web is given by the equation \(z = x + y\). Its automorphisms have the form \(x = a\tilde {x} + {{b}_{1}}\), \(y = a\tilde {y} + {{b}_{2}}\), and the corresponding autotopies \(({{A}_{1}},{{A}_{2}},{{A}_{3}})\) of a three-web (that is, admissible transformations of the parameters of the families forming the web) are \(x = a\tilde {x} + {{b}_{1}}\), \(y = a\tilde {y} + {{b}_{2}}\), \(z = a\tilde {z} + {{b}_{1}} + {{b}_{2}}\). It follows that any regular web (that is, a web locally diffeomorphic to a parallel one) also admits a three-parameter group of automorphisms. Therefore, webs that admit smaller families of automorphisms are of interest.
Three-webs with automorphisms have been studied by many authors, starting with Cartan [1]. We do review this topic and just mention two of the latest works in this direction [2, 3].
The following statements were proven in [4].
Theorem 1. If a curvilinear three-web on the real plane admits a one-parameter family of automorphisms (\(AW\)-webs), then in some local coordinates its equation can be reduced to the form
Here, \(\lambda (x - y)\) is an arbitrary smooth function of the variable \(x - y\).
Theorem 2. If a curvilinear three-web on the real plane admits a two-parameter family of automorphisms, then it is regular.
The proofs in [4] were based on the simple fact that the absolute invariants of a web are constant along the trajectories of automorphisms. In the current paper, we prove Theorem 1 in a different way, directly integrating the corresponding system of differential equations, and also show that the trajectories of automorphisms of an \(AW\)-web are geodesics of the Chern connection of a three-web. In addition, in this paper we found all \(AW\)-webs for which one of the covariant derivatives (with respect to the canonical Chern connection of this web) is equal to zero.
2. Let \(W\) be an arbitrary curvilinear three-web formed in a certain region \(D\) of the plane by lines \(x = \operatorname{const} \), \(y = \operatorname{const} \), and \(f(x,y) = \operatorname{const} \), then the equation of this web has the form \(z = f(x,y).\) We put
By performing exterior differentiation of the forms \({{\omega }_{1}}\) and \({{\omega }_{2}}\), we get
where
Putting \(\omega = \Gamma ({{\omega }_{1}} + {{\omega }_{2}}),\) we rewrite equalities (3) as
Exterior differentiation of the form \(\omega \) leads to the equality
where
Equations (4) and (5) are called structure equations of a three-web \(W\), and the function \(b\) is referred to as the curvature of this web. The condition \(b = 0\) characterizes the class of regular webs.
On the other hand, Eqs. (4) and (5) are equations of some torsion-free affine connection, and the forms of connection are given by
This connection is called the Chern connection.
The geodesic lines of the Chern connection are given by the equations
where \(d\) is the ordinary differentiation and \(\Theta \) is some 1-form depending on the choice of parameter on the geodesic. Excluding \(\Theta \) from Eqs. (6), we arrive at the geodesic equation in another form:
From here we get \({{\omega }_{1}} = C{{\omega }_{2}},\) \(C = \operatorname{const} ,\) or
At \(C = 0,\infty , - 1\) we get lines of the first, second, and third families of the web \(W\). At \(C = 1\) we obtain a family of geodesic lines, which, together with the lines of the third family, harmonically divide (at each point) a pair of lines of the first and second families. We can also say that the family \(C = 1\) is associated with the third family of web lines relative to the pair of the first two. Therefore, we call this family of lines 3-conjugate.
3. Automorphism of a three-web \(W\) is a local diffeomorphism of a domain \(D\), which translates the lines of the web \(W\) again into the line of this web. The proof of the following statement can be found in ([6], Theorem 6.8). Local automorphisms of a three-web \(W\) are also automorphisms of the corresponding Chern connection. Conversely, let \(\varphi \) be an automorphism of the Chern connection of some three-web \(W\), defined in the domain \(D\), and there exists a point \(p\) in \(D\) such that the differential \({{\left. {d\varphi } \right|}_{p}}\) translates tangents to lines of the web \(W\) passing through the point \(p\) into tangents to the corresponding lines passing through the point \(\varphi (p)\). Then \(\varphi \) is an automorphism of the three-web \(W\).
(Recall that the automorphism of an affine connection \(\Gamma \) defined on a manifold \(X\) is a diffeomorphism of this manifold that preserves the law of parallel translation, that is, preserves the covariant differential with respect to this connection.)
Proof of Theorem 1. Suppose that the three-web \(W\) admits a one-parameter family of automorphisms defined by a vector field \(\xi ({{\xi }_{1}},{{\xi }_{2}})\). Then the quantities \({{\xi }_{1}},{{\xi }_{2}}\) satisfy the following equations from [5]:
or, by virtue of (2) and (3),
Hence,
We put \({{\xi }_{1}} = \alpha {{f}_{x}}\), \({{\xi }_{2}} = \beta {{f}_{y}}\). Substituting into the previous equations, we get
This implies \(\alpha = \alpha (x)\), \(\beta = \beta (y)\) and
Lemma. By variable replacements \(x = x(\tilde {x})\), \(y = y(\tilde {y})\) the functions \(\alpha (x)\) and \(\beta (y)\) can be reduced to unity.
Proof. Let us denote \(f(x(\tilde {x}),y(\tilde {y})) = \tilde {f}(\tilde {x},\tilde {y})\), then
We put \(dx{\text{/}}d\tilde {x} = \alpha (x)\), \(dy{\text{/}}d\tilde {y} = \beta (y)\) and substitute it into (8). After transformations we arrive at the equation
Let us continue the proof, assuming that the indicated change of variables has been made and omit the tilde over the variables. After some calculations, we reduce Eq. (9) to the form
which leads to
where \(\varphi (x - y)\) is a smooth function of the variable \(x - y\). Let us further put \(f(x,y)\) = \(g(u,{v})\), \(u = x + y\), \({v} = x - y\), then Eq. (10) becomes
or \({{g}_{{v}}} = {{g}_{u}}\nu ({v}).\) We have
As a result, the web equation \(z = f(x,y)\) takes the form \(z = g(u + \lambda ({v}))\) = \(f(x + y + \lambda (x - y)\). After admissible parameter replacement \({{f}^{{ - 1}}}(z) \to z\) we arrive at Eq. (1).
\(\square \)
Because \(\alpha = \beta = 1\), it is true that \(\xi = ({{\xi }_{1}},{{\xi }_{2}}) = ({{f}_{x}},{{f}_{y}})\). Automorphisms of the \(AW\)-web have the form \(x \to x + a\), \(y \to y + a\). The trajectories of the automorphisms are the lines \(x - y = {\text{const}}\).
Theorem 3. Trajectories of automorphisms of the \(AW\)-web are geodesic and coincide with its \(3\)-conjugate family only if the three-web is regular.
Proof. From Eq. (1) we find \({{f}_{x}} = 1 + \lambda {\kern 1pt} '(x - y)\), \({{f}_{y}} = 1 - \lambda {\kern 1pt} '(x - y)\), so the geodesic equation (7) for a three-web \(AW\) becomes
where prime means the derivative with respect to the variable \({v} = x - y\). On the trajectory of automorphisms \(x - y = \operatorname{const} \) we have \(\lambda {\kern 1pt} '(x - y) = \operatorname{const} \). We can choose the constant \(C\) so that the geodesic equation takes the form \(dx - dy = 0\) or \(x - y = {\text{const}}\). The first part of Theorem 3 is proven.
The equation of the 3-conjugate family for the \(AW\)-web transforms to
This equation coincides with the equation for the trajectories of automorphism \(dx - dy = 0\) only in case \(\lambda {\kern 1pt} ' = 0\) or \(\lambda = {\text{const}}\). But then Eq. (1) defines a regular three-web.
\(\square \)
4. Let us find the covariant derivatives of the curvature of web (1), denoting it, as above, by \(AW\). First, we calculate the curvature \(b\). We have
Further,
We find the covariant derivatives of the curvature \(b\) from the formula
which is a differential continuation of the structure equation (5). We have
Consequently,
5. Let us find three-webs \(AW\) for which \({{b}_{1}} = 0\). According to ([7], p. 69) this condition distinguishes three-webs \({{B}_{1}}\) formed by a family of parallel lines and integral curves of two Riccati equations of a special form. Let us show that such a class exists, and the solution can be found in quadratures. Note that, to describe this class, we cannot use the results from [7], because the moving frame there was normalized by the condition that one of the covariant derivatives of the curvature is equal to unity, while in this article the normalization is different—the trajectories of automorphisms are written in the form \(x - y = {\text{const}}\).
As can be seen from (12), the condition \({{b}_{1}} = 0\) leads to the equation
or \((\ln b){\kern 1pt} ' = (\ln {{(1 - \lambda {\kern 1pt} ')}^{{ - 2}}}){\kern 1pt} '\). Hence,
If \(\alpha = 0\), then \(b = 0\), and we get a regular web. Let further \(\alpha \ne 0\). Comparing the last equality with (11), we arrive at the relation
or
where
In (13) we put
As a result, (13) takes the form \(2\Theta d\Theta = \alpha {{e}^{t}}dt.\) Therefore, \({{\Theta }^{2}} = \alpha {{e}^{t}} + \beta \), \(\beta = \operatorname{const} ,\) and
Replacement \(\alpha {{e}^{t}} + \beta = {{u}^{2}}\) reduces the integral to the form
Case 1. \(\beta = {{\delta }^{2}} \ne 0\), then
After simple transformations, \(\alpha > 0\) get for
for \(\alpha < 0\)
As we can see, these cases are identical, so we consider only the first option below. It follows from (14) that
Substituting here the previous expression for \({{e}^{t}}\), we find
Next, we put \(\frac{{\delta {v} + \gamma }}{2} = X\) and in the right-hand side of (16) we express \({{\sinh }^{2}}X\) through \(\tanh X\). As a result, we get
After standard replacement \(\tanh X = z\) we arrive at the integral
Subcase 1a. \(\frac{\varepsilon }{{\varepsilon - 1}} = {{\eta }^{2}}\), and \(\eta \ne 0\), because \(\varepsilon = {{\delta }^{2}}{\text{/}}\alpha \ne 0\). Then we have
Subcase 1b. At \(\frac{\varepsilon }{{\varepsilon - 1}} = - {{\eta }^{2}}\) from (17) we obtain
Case 2. \(\beta = - {{\delta }^{2}} \ne 0\). Then it follows from (15) that
which leads to
Putting \(\frac{{\delta {v} + \gamma }}{2} = X\) and \(\frac{{{{\delta }^{2}}}}{\alpha } = \eta \), we have
Subcase 2a. \((\eta + 1){\text{/}}\eta = - {{\kappa }^{2}} \ne 0\) \( \Rightarrow \)
We put \(\tan X = z\), then
However,
Hence,
Subcase 2b. \((\eta + 1){\text{/}}\eta = {{\kappa }^{2}} \ne 0\) \( \Rightarrow \)
which results in
However,
Therefore,
Subcase 2c. \(\kappa = 0\). Then
and
Case 3. \(\beta = 0\). From (15) we find
This leads to
Integration yields
for \(\alpha > 0\)
for \(\alpha < 0\)
We have proved the following theorem.
Theorem 4. There are seven types of curvilinear three-webs admitting a one-parameter family of infinitesimal automorphisms for which one of the covariant derivatives of the curvature is equal to zero. These are three-webs defined by equations of the form (1), where the function \(\lambda \) is calculated by formulas (18)–(24).
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Shelekhov, A.M. Сurvilinear Three-Webs with Automorphisms. Russ Math. 68, 77–84 (2024). https://doi.org/10.3103/S1066369X24700464
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DOI: https://doi.org/10.3103/S1066369X24700464