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\begin{document}
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{\sf Uzbek Mathematical}\\
{\sf Journal, 2019,\ No 2, pp.\pageref{firstpage}-\pageref{lastpage}}\\
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\begin{center}
\textbf{\large Second-order differential invariants of submersions}\\
\textbf{ X.F.Sharipov}
\end{center}
{\small \textbf{Abstract.} In this paper, we study the second-order
differential invariants of submersions with respect to the group of
conformal transformations. In particular, it is proved that the
ratio of principal surface curvatures is a second-order differential
invariant with respect to the group of conformal transformations.
\\
\textbf{Keywords:} Conformal transformation, differential
invariants, submersion, vector field.
\textbf{MSC (2010):} 53C12, 57R25, 57R35
\\
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{\it Second-order differential invariants of submersions }\hfill
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\section{Introduction}
\ \ \ \ Felix Klein in the Erlangen program proposed a unified approach to the description of various geometries. According to this program, one of the main problems of geometry is the construction of invariants of geometric objects with respect to the action of the group defining this geometry. The construction of invariants is based on the ideas of Sophus Lie, who introduced Lie groups into the geometry. In particular, when considering classification problems and equivalence problems in differential geometry, one should consider differential invariants with respect to the action of Lie groups. In this case, the problem of equivalence of geometric objects is reduced to finding the complete system of differential invariants \cite{Kleyn}.
The definition of a differential invariant of $k-$ th order as a function on the space $k-$ of jets of sections of the corresponding bundle made it possible to work effectively with them, and using invariant differentiation, new invariants can be obtained from known differential invariants.
Depending on the geometry, the orders of the first non-trivial differential invariants can be different. For example, in the space $ R^3 $ equipped with the Euclidean metric, the complete system of differential invariants of a curve is its curvature and torsion, which are second and third order invariants, respectively. The first differential invariant of a curve with respect to projective transformations is of the seventh order.
In this paper, we study the second-order differential invariants of submersions with respect to the group of conformal transformations.
The study of differential invariants of submersions has been the subject of numerous studies \cite{Alekseyevskiy,Kushner,Kuzakon1,Kuzakon2,Narmanov,st3,st4}. In \cite{st3}, second and third order differential invariants of submersions are found with respect to conform transformations.
Second-order differential invariants of submersions of Euclidean spaces with respect to the group of motions are studied in the papers \cite{Kuzakon1,Kuzakon2}.
\section{Preliminaries}
\ \ \ \ Let $ M- $ be a smooth Riemannian manifold of dimension $ n, $ $ B $ a smooth Riemannian manifold of dimension $ m, $ $ n> m. $
\begin{defn} \label{defn1} Differentiable mapping $\pi:M\to B $ of maximal rank is
called submersion.
\end{defn}
Let $ G $ be Lie group of transformation of Riemannian manifold $ M. $ If the group $ G $ is a $ k- $ dimensional Lie group, then it has $ k $ infinitesimal generators (vector fields).
\begin{defn} \label{defn2} The function $ I (p) $ on $ M $ is called the invariant of the transformation group $ G, $ if $ I (p) = I (gp) $ for each element of $ g \in G, $
$ p \in M. $
\end{defn}
Let $ M, B $ be smooth manifolds and $p\in M.$ Suppose $ f, g: M \rightarrow B $ are smooth maps with $ f (p) = g (p) = q. $
1) $ f $ has a first order contact with $ g $ at a point $ p $ if $ (df)_p = (dg)_p $ as mapping of $ T_pM \rightarrow T_qB. $
2) $ f $ has $k-$th order contact with $ g $ at a point $ p $ if $ (df): TM \rightarrow TB $ has $(k-1)$st order contact with $(dg)$ at every point in $T_pM.$ This is written as $ f \sim_k g $ at $ p $ ($ k$ is a positive integer).
Denote by $ J^k (M, B)_{p, q} $ - sets of equivalence classes under "$ \sim_k $ at $ p $"\ of mappings $ f: M \rightarrow B$ where $f(p) = q. $
We put $ J^k(M,B)=\bigcup_{(p,q)\in{M \times B}} J^k(M,B)_{p,q}. $ It is known that this set is a smooth manifold of dimension $ n+m \sum_{i = 0}^kC_{n+i-1} ^i $ \cite{Vinogradov}.
\begin{defn} \label{defn3}
The manifold $ J^k(M, B) $ is called the space of $k-$ jets.
\end{defn}
The action of the group $ G $ on $ M $ generates some action of the group on $ J^k(M, B). $ This action is called the $ k- $ th prolongation of the action of the group $ G $ on $ J^k (M, B).$ The infinitesimal generators of the $ k- $ th prolongation of the group $ G $ on $ J^k (M, B) $ are the $ k- $ th prolongations of the infinitesimal generators of the group $ G. $
\begin{defn} \label{defn4} The function $ I \in J^k (M, B) $ is called a differential invariant of order $ k $ of the group $ G, $ if it is preserved under the action of the $ k- $ th prolongation $ G $ on $ J^k (M, B). $
\end{defn}
The so-called contact vector fields play an important role in finding the differential invariants.
As known \cite{Alekseyevskiy}, the contact vector field with the generating function $ f $ has the form
$$X_f=-\sum_{i=1}^n\frac{\partial f}{\partial p_i}\frac{\partial }{\partial x_i}+(f-\sum_{i=1}^np_i\frac{\partial f}{\partial p_i})\frac{\partial}{\partial u}+\sum_{i=1}^n(\frac{\partial f}{\partial x_i}+p_i\frac{\partial f}{\partial u})\frac{\partial }{\partial p_i}$$
Accordingly, the prolongation of the vector field $X_f$ in $J^2(R^n,R)$ is
$$X_f^{(2)}=X_f+\sum_{i\leq j}f_{ij}\frac{\partial}{\partial p_{ij}}$$
where
$$f_{ij}=\frac{\partial^2 f}{\partial x_i \partial x_j}+p_i\frac{\partial^2 f}{\partial x_j \partial u}+p_j\frac{\partial^2 f}{\partial x_i \partial u}+p_ip_j\frac{\partial^2 f}{\partial u^2}+p_{ij}\frac{\partial f}{\partial u}+$$
$$+\sum_{r=1}^n\left(p_{rj}\frac{\partial^2 f}{\partial x_i \partial p_r}+p_{ri}\frac{\partial^2 f}{\partial x_j \partial p_r}+p_ip_{rj}\frac{\partial^2 f}{\partial u \partial p_r}+p_jp_{ri}\frac{\partial^2 f}{\partial u \partial p_r}\right)+
\sum_{r,s=1}^np_{ir}p_{js}\frac{\partial^2 f}{\partial p_r \partial p_s}$$
\begin{ex} Consider the submersion $ \varphi: R^{1,2}
\rightarrow R, $ where $ R^{1,2} $ is the two-dimensional Minkowski
plane. Recall that in this case the metric has the form
$$ds^2=-(dx_1)^2+(dx_2)^2$$
Consider the group $ G, $ transformations of the Minkowski plane, generated by the flows of vector fields
$$X_{i}=\frac{\partial}{\partial x_i}, i=1,2, X=x_2\frac{\partial}{\partial x_1}+x_1\frac{\partial}{\partial x_2}, $$
and the group of diffeomorphisms of the line generated by the vector field
$$ Y=u\frac{\partial}{\partial u}.$$
The prolongations of these vector fields to $J^2(R^{1,2},R)$ have the following types, respectively:
$$X_{p_i}^2=\frac{\partial}{\partial x_i}, i=1,2,$$
$$X_{x_2p_1+x_1p_2}^2=x_2\frac{\partial}{\partial x_1}+x_1\frac{\partial}{\partial x_2}+p_2\frac{\partial}{\partial p_1}+p_1\frac{\partial}{\partial p_2}+(p_{11}+p_{22})\frac{\partial}{\partial p_{12}}+2p_{12}\frac{\partial}{\partial p_{22}}$$
$$Y_{h(u)}^2=u\frac{\partial}{\partial u}+\sum_{i=1}^2p_i\frac{\partial}{\partial p_i}+\sum_{1\leq i\leq j\leq 2}p_{ij}\frac{\partial}{\partial p_{ij}}.$$
Differential second order invariants of submersion $ \varphi: R^{1,2} \rightarrow R $ are the following functions:
$$I_1=\frac{p_2^2p_{11}+2p_1p_2p_{12}+p_1^2p_{22}}{(p_2^2-p_1^2)^{\frac{3}{2}}},$$
$$I_2=\frac{(p_1^2+p_2^2)p_{12}-p_1p_2(p_{11}+p_{22})}{(p_2^2-p_1^2)^{\frac{3}{2}}}.$$
\end{ex}
\section{Main Part}
\ \ \ \ Now consider a conformal vector field in three-dimensional Euclidean space:
\begin{equation}\label{сс}
X=x_1\frac{\partial}{\partial x_1}+x_2\frac{\partial}{\partial x_2}+x_3\frac{\partial}{\partial x_3}
\end{equation}
The flow of this vector field generates conformal mappings. Let us find invariant functions and invariant sets of groups of these mappings.
The following theorem \cite{Alekseyevskiy} is known.
\begin{thm} \label{thm1} The function $ I $ is an invariant of order $ k $ of a
transformation group $ G $ only and only if it is the first integral
of the infinitesimal generator of $ G^k. $
\end{thm}
By this theorem, the function $ F: R^{3} \rightarrow R $ is an invariant if and only if
$$X(F)=x_1\frac{\partial F}{\partial x_1}+x_2\frac{\partial F}{\partial x_2}+x_3\frac{\partial F}{\partial x_3}=0.$$
From here we find that the following functions are invariant functions:
$$F_1=\frac{x_1}{x_2}, F_2=\frac{x_2}{x_3}.$$
We have got two invariant functions. The level surfaces of these functions are planes that pass through the axes of coordinates and therefore, when conformally transformed, transforms into themselves.
Let $ \varphi: R^3 \rightarrow R- $ be a submersion. It is known that the level surfaces of the submersion are regular surfaces.
Let us to find the invariants of this submersion (invariants of level surfaces) with respect to the conformal transformations generated by the flow of a vector field (3.1).
The generating function of this conformal vector field is $$X(\varphi)=x_1\frac{\partial\varphi}{\partial x_1}+x_2\frac{\partial\varphi}{\partial x_2}+x_3\frac{\partial\varphi}{\partial x_3}=x_1p_1+x_2p_2+x_3p_3.$$
The contact vector field with the generating function $ f = x_1p_1 + x_2p_2 + x_3p_3 $ has the following form:
$$X^1_f=-x_1\frac{\partial}{\partial x_1}-x_2\frac{\partial}{\partial x_2}-x_3\frac{\partial}{\partial x_3}+
p_1\frac{\partial}{\partial p_1}+p_2\frac{\partial}{\partial p_2}+p_3\frac{\partial}{\partial p_3}$$
Find the first order differential invariant of this submersion: function $I$ is first order differential invariant of this submersion if and only if
$$X^1_f(I)=0.$$
From here we find two invariants: $$ I_1 = \frac{p_1^2 + p_3^2}{p_1p_2}, I_2 = \frac{p_2^2 + p_3^2}{p_1p_2} $$
Really,
$$X^1_f(I_1)=p_1\frac{p_1^2-p_3^2}{p_1^2p_2}-p_2\frac{p_1^2+p_3^2}{p_1p_2^2}+p_3\frac{2p_3}{p_1p_2}=
\frac{p_1^2-p_3^2}{p_1p_2}-\frac{p_1^2+p_3^2}{p_1p_2}+\frac{2p_3^2}{p_1p_2}=0$$
$$X^1_f(I_2)=-p_1\frac{p_2^2+p_3^2}{p_1^2p_2}+p_2\frac{p_2^2-p_3^2}{p_1p_2^2}+p_3\frac{2p_3}{p_1p_2}=
-\frac{p_2^2+p_3^2}{p_1p_2}+\frac{p_2^2-p_3^2}{p_1p_2}+\frac{2p_3^2}{p_1p_2}=0$$
The prolongations of the vector field $ X $ to $ J^2 (R^3, R) $ have the form
$$X^2_f=X^1_f+2p_{11}\frac{\partial}{\partial p_{11}}+2p_{12}\frac{\partial}{\partial p_{12}}+2p_{13}\frac{\partial}{\partial p_{13}}+2p_{22}\frac{\partial}{\partial p_{22}}+2p_{23}\frac{\partial}{\partial p_{23}}+2p_{33}\frac{\partial}{\partial p_{33}}.$$
Let us to find the second-order differential invariant of this submersion. Function $I$ is second order differential invariant of this submersion if and only if
$$ X^2_f (I) = 0. $$
From here we find the following invariants: $$ I_1 = \frac{\left|
\begin{array}{cc}
p_1 & p_2 \\
p_{11} & p_{12} \\
\end{array}
\right|}{\left|
\begin{array}{cc}
p_1 & p_2 \\
p_{12} & p_{22} \\
\end{array}
\right|}, I_2 = \frac{\left|
\begin{array}{cc}
p_1 & p_2 \\
p_{12} & p_{22} \\
\end{array}
\right|}{\left|
\begin{array}{cc}
p_1 & p_2 \\
p_{13} & p_{23} \\
\end{array}
\right|}, $$ $$ I_3 = \frac{\left|
\begin{array}{cc}
p_2 & p_3 \\
p_{12} & p_{13} \\
\end{array}
\right|}{\left|
\begin{array}{cc}
p_2 & p_3 \\
p_{22} & p_{23} \\
\end{array}
\right|}, I_4 = \frac{\left|
\begin{array}{cc}
p_2 & p_3 \\
p_{22} & p_{23} \\
\end{array}
\right|}{\left|
\begin{array}{cc}
p_2 & p_3 \\
p_{23} & p_{33} \\
\end{array}
\right|}. $$
Now consider the submersion $ \varphi: R^3 \rightarrow R, \varphi(x_1, x_2, x_3) = f(x_1, x_2) -x_3. $ The level surfaces of this submersion are regular surfaces given by the explicit function $ L_c = \{(x_1, x_2, x_3) \in R^3: x_3 = f (x_1, x_2) -c \}. $
Let us to find the second-order differential invariants of this submersion (invariants of level surface) with respect to conformal transformations generated by the flow of a vector field (3.1). To do this, we find the prolongation of the flow of a conformal vector field (3.1) in $ J^2 (x_1, x_2, x_3, p_1, p_2, p_ {11}, p_ {12}, p_ {22}),$ where $ p_1 = \frac{\partial x_3} {\partial x_1}, $ $ p_2 = \frac {\partial x_3} {\partial x_2}, $ $ p_ {11} = \frac {\partial^2 x_3} {\partial x_1^2 }, $ $ p_ {12} = \frac {\partial^2 x_3} {\partial x_1 \partial x_2}, $ $ p_ {22} = \frac {\partial^2 x_3} {\partial x_2^2 }. $
Let the flow of a vector field (3.1) translates the point $ (x_{1},
x_{2}, x_{3}), $ to the point $ (x'_{1}, x'_{2}, x'_{3}). $ Let us
find the transformation formulas for the derivatives under this
conformal transformation.
Now we find the first partial derivatives.
\begin{equation*}
\frac{\partial x'_3}{\partial x'_i}=\frac{\partial e^tx_3}{\partial e^tx_i}=\frac{\partial x_3}{\partial x_i}=p_i,i=1,2
\end{equation*}
and second partial derivatives
\begin{equation*}
\frac{\partial^2 x'_3}{\partial x'_i\partial x'_j}=\frac{\partial }{\partial x'_i}\frac{\partial x'_3}{\partial x'_j}=\frac{\partial }{\partial e^tx'_i}\frac{\partial e^tx'_3}{\partial e^tx'_j}=e^{-t}\frac{\partial^2 x_3}{\partial x_i\partial x_j}=e^{-t}p_{ij},i,j=1,2.
\end{equation*}
The prolongation of a vector field (3.1) in $J^2(x_1,x_2,x_3,p_1,p_2,p_{11},p_{12},p_{22}),$ has the following form:
\begin{equation}\label{c}
X^2=x_1\frac{\partial}{\partial x_1}+x_2\frac{\partial}{\partial x_2}+x_3\frac{\partial}{\partial x_3}-p_{11}\frac{\partial}{\partial p_{11}}-p_{12}\frac{\partial}{\partial p_{12}}-p_{22}\frac{\partial}{\partial p_{22}}
\end{equation}
Consequently, the flow of the vector field (3.2) has the following form:
$$(x_1,x_2,x_3,p_1,p_2,p_{11},p_{12},p_{22})\rightarrow(e^tx_1,e^tx_2,e^tx_3,p_1,p_2,e^{-t}p_{11},e^{-t}p_{12},e^{-t}p_{22}).$$
The directions in the tangent plane where the curvature takes its maximum and minimum values are always perpendicular, if $k_1$ does not equal $k_2$, and are called principal directions.
\begin{thm} \label{th2}Under the transformations generated by the
flow of the vector field (3.1), the principal directions of the
level surfaces of the submersion are transferred to the principal
directions.
\end{thm}
\noindent{\bf Proof.} Let the tangent vector $
\overrightarrow{a}=\overrightarrow{r_{x_{1}}}a_1+\overrightarrow{r_{x_{2}}}a_2$
define one of the principal directions at $ (x_{1}, x_{2}, x_{3}). $
To determine the principal direction, it suffices to find the
relation $ \lambda = \frac{a_1}{a_2}. $ This relation is determined
from the following quadratic equation:
\begin{equation}\label{сс}
\left(LF-ME\right)\lambda^2+(GL-EN)\lambda+MG-FN=0,\end{equation}
where $ E, F, G- $ are the coefficients of the first quadratic form and $ L, M, N- $ are the coefficients of the second quadratic form.
The coefficients of the first and second quadratic forms are calculated by the formulas
\begin{equation*}
E=1+p_1^2,F=p_1p_2, G=1+p_2^2,
\end{equation*}
\begin{equation*}
L=\frac{p_{11}}{\left(1+p_1^2+p_2^2\right)^{\frac{1}{2}}}, M=\frac{p_{12}}{\left(1+p_1^2+p_2^2\right)^{\frac{1}{2}}}, N=\frac{p_{22}}{\left(1+p_1^2+p_2^2\right)^{\frac{1}{2}}}.
\end{equation*}
Therefore, equation (3.3) has the following form
\begin{equation*}
\left(p_{11}p_1p_2-p_{12}(1+p_1^2)\right)\lambda^2+\left(p_{11}(1+p_2^2)-p_{22}(1+p_1^2)\right)\lambda+
\end{equation*}
\begin{equation*}
+p_{12}(1+p_2^2)-p_{22}p_1p_2=0.
\end{equation*}
This equation goes into equivalent equation under the conformal
transformations generated by the flow of a vector field (3.1),
therefore the principal direction is transferred to the principal
direction. The theorem is proved.
\hfill$\Box$ \
\begin{thm} \label{thm3} The ratio $ \frac{k_{_{1}}}{k_{2}}$ of principal
curvatures of level surface of a submersion is second-order
differential invariant of the group of conformal transformations
generated by the flow of a vector field (3.1).
\end{thm}
\noindent{\bf Proof.} By definition, the principal curvature is the
normal curvature in the principal direction.
Let $\overrightarrow{a}=(a_1,a_2),\overrightarrow{b}=(b_1,b_2)-$ principal directions, then for the corresponding principal curvatures it takes following equalities:
\begin{equation*}
k_1=\frac{II(\overrightarrow{a})}{I(\overrightarrow{a})},k_2=\frac{II(\overrightarrow{b})}{I(\overrightarrow{b})}.
\end{equation*}
By theorem \ref{th2}, the principal direction is transferred to the
principal direction under the conformal transformation generated by
the vector field flow (3.1). From the formula for the coefficients
of the first and second quadratic forms, it follows that the for the
principal curvatures of the image of a level surface under conformal
transformation generated by the flow of a vector field (3.1) it
takes following equalities:
\begin{equation*}
k'_1=e^{-t}\frac{II(\overrightarrow{a})}{I(\overrightarrow{a})}=e^{-t}k_1,k'_2=e^{-t}\frac{II(\overrightarrow{b})}{I(\overrightarrow{b})}=e^{-t}k_2.
\end{equation*}
Therefore, ratio $ \frac{k_{_{1}}}{k_{2}}$ of principal curvatures
of level surface of the submersion do not change under conformal
transformations generated by the flow of the vector field (3.1). The
theorem is proved.
\hfill$\Box$ \
\begin{ex} Consider the submersion $ \varphi: R^3
\rightarrow R, $ defined by the formula
\begin{equation}\label{сс}
\varphi(x_1,x_2,x_3)=\frac{1}{2}\left(x_1^2+x_2^2\right)-x_3.
\end{equation}
The level surfaces of this submersion are elliptic paraboloids.
The principal directions of an elliptic paraboloid are found using a quadratic equation: $$
x_1x_2\lambda^2+(x_2^2-x_1^2)\lambda-x_1x_2=0.$$ From here we find the principal directions of the elliptic paraboloid $\overrightarrow{a}=(-x_2,x_1)$ and $\overrightarrow{b}=(x_1,x_2),$ they are known to be orthogonal.
The principal curvatures of an elliptic paraboloid are calculated by the formulas
\begin{equation}\label{сс}
k_1=\frac{1}{(1+x_1^2+x_2^2)^{\frac{1}{2}}},k_2=\frac{1}{(1+x_1^2+x_2^2)^{\frac{3}{2}}}.
\end{equation}
Their ratio $ \frac{k_1}{k_2} = 1 + x_1^2 + x_2^2 $ does not change under the conformal transformations generated by the flow of the vector field (3.1).
\end{ex}
Recall that the direction at a point on a surface is called asymptotic if the normal curvature in this direction is zero.
\begin{thm} \label{thm4} Under conformal transformations generated by the
flow of the vector field (3.1), the asymptotic direction of the
level surfaces of the submersion is transferred to the asymptotic
direction .
\end{thm}
\noindent{\bf Proof.} Let the tangent vector $
\overrightarrow{a}=\overrightarrow{r_{x_{1}}}a_1+\overrightarrow{r_{x_{2}}}a_2$
determine the asymptotic direction at the point
$(x_{1},x_{2},x_{3}).$
To determine the asymptotic direction, it is sufficiently to find the ratio $\lambda=\frac{a_1}{a_2}.$
This ratio is determined from the following quadratic equation:
\begin{equation}\label{сс}
L\lambda^2+2M\lambda+N=0,\end{equation}
where $L,M,N-$ are the coefficients of the second quadratic form.
The coefficients of the second quadratic form are calculated by the formulas
\begin{equation*}
L=\frac{p_{11}}{\left(1+p_1^2+p_2^2\right)^{\frac{1}{2}}}, M=\frac{p_{12}}{\left(1+p_1^2=p_2^2\right)^{\frac{1}{2}}}, N=\frac{p_{22}}{\left(1+p_1^2+p_2^2\right)^{\frac{1}{2}}}.
\end{equation*}
Therefore, equation (3.6) has the following form
\begin{equation*}\label{сс}
p_{11}\lambda^2+2p_{12}\lambda+p_{22}=0,\end{equation*}
This equation goes into equivalent equation under the conformal
transformations generated by the flow of a vector field (3.1),
therefore the asymptotic direction is transferred to the asymptotic
direction. The theorem is proved.
\hfill$\Box$ \
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\bibitem{Vinogradov} Vinogradov A.M., Krasilshchik I.S., Lychagin V.V. Introduction to the geometry of nonlinear differential equations. M. Science. Ch. ed.
Phys.-Mat. lit., 1986. 336 p.
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Sharipov X.F.,\\
National University of Uzbekistan named after Mirzo Ulugbek, Tashkent 100174, Uzbekistan. \\
e-mail: sh\_xurshid@yahoo.com\\
\\
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