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section{introduction} the concept of {sl cartan geometry} appeared at the beginning of the twentieth century, when {e}lie cartan was working on the so-called {sl equivalence problem}, the aim of which is to determine whether two given geometric structures can be mapped bijectively onto each other by some diffeomorphism. this problem can be considered in many different contexts, such as equivalences of submanifolds, of differential equations, of frames, of coframes and of other geometric structures. in the specific case of local real analytic hypersurfaces in $bbb c^2$, poincare (1907) initiated the study of the {sl cauchy-riemann} (cr for short) equivalence problem under biholomorphic transformations. later, in 1932, this problem was solved in an essentially complete way by cartan cite{cartan}. in general, cartan also developed appropriate concepts and showed that one can reformulate several,,---,,somewhat hard,,---,,initial equivalence questions ({em see}~cite{olver1}) in terms of equivalences of coframes. granted this, he devised an algorithm to decide whether two given manifolds $m_1$ and $m_2$ equipped with certain specific geometric structures encoded by means of coframes are equivalent. the main thrust is to construct two principal bundles ${cal g}_1$ and ${cal g}_2$ over $m_1$ and $m_2$ having the same structure group together with two coframes $omega^1:={omega_1^1,ldots,omega_1^n}$ on ${cal g}_1$ and $omega^2:={omega_2^1,ldots,omega_2^n}$ on ${cal g}_2$, such that $m_1$ and $m_2$ are equivalent if and only if there exists a diffeomorphism $phi:{cal g}_1 o {cal g}_2$ commuting with projections, which sends $omega^1$ to $omega^2$, i.e.: [ phi^ast(omega_2^i) = omega_1^i, scriptstyle{(i,=,1,,ldots,,n)}. ] this also motivated cartan to introduce new elegant geometries, that he called {sl espaces g{e}n{e}ralis{e}s} and that are nowadays defined as follows. egin{definition} label{cartanconnecdefini} let $g$ be a lie group with a closed subgroup $h$, and let $frak g$ and $frak h$ be the corresponding lie algebras. a {sl cartan geometry of type $(g,h)$} on a manifold $m$ is a principal $h$-bundle: [ pi:{cal g}longrightarrow m ] together with a $frak g$-valued $1$-form $omega$, called the corresponding {sl cartan connection}, on $cal g$ subjected to the following three conditions: smallskipegin{itemize} item[ extbf{(i)}] $omega_p:t_p{cal g}longrightarrowfrak g$ is an isomorphism at every point $pincal g$; smallskip item[ extbf{(ii)}] if $r_h(p):=ph$ is the right translation on $cal g$ by $hin h$, then for any such $h$: [ r^ast_homega={ m ad}(h^{-1})circomega; ] smallskipitem[ extbf{(iii)}] $omega(h^dag)={sf h}$ for every ${sf h}infrak h$, where: [ h^dag|_p := extstyle{frac{d}{dt}}ig|_0ig((r_{exp(t sf h)}(p)ig) ] is the left-invariant vector field on $cal g$ corresponding to $sf h$. end{itemize} oindent end{definition} underlying a cartan geometry, there always is a homogeneous space, namely $g/h$. in fact, among the cartan geometries of type $(g,h)$, the most symmetric one, called {sl klein geometry of type $(g,h)$}, arises when $m=g/h$, when $pi colon g ightarrow g/h$ is the projection onto left-cosets, and when $omega=omega_{mc} colon tg o frak g$ is the {sl maurer-cartan form} on $g$ ({em see} cite{sharpe}). generally, cartan geometries are a generalization of klein geometries and also, are a generalization of {sl riemannian geometries}. while the geometries of klein present perfect homogeneity and while the ones of riemann can be regarded as inhomogeneous types of euclidean geometry, cartan devised a broad synthesis between these two seemingly incompatible types of geometry. in general, with a cartan connection $omega$ as above, if we associate the vector field $widehat{x}:=omega^{-1}({sf x})$ on $cal g$ to an arbitrary element ${sf x}$ of $frak g$, then the infinitesimal version of condition extbf{(ii)} reads as: [ [widehat{x},widehat{y}] = widehat{[{sf x},{sf y}]_{frak g}}, ] whenever {sf y} belongs to $frak h$. but in the special case of klein geometries, this equality holds moreover for any arbitrary element $sf y$ of $frak g$. this difference motivates one to define the {sl curvature function}: [ kappa colon {cal g} longrightarrow { m hom} ig( lambda^2 (frak g/frak h),frak g ig) ] associated to the cartan connection $omega$ by: [ kappa_p({sf x},{sf y}) := omega_pig([widehat{x}, widehat{y}]ig)-[{sf x},{sf y}]_{frak g} {scriptstyle{(p,in,mathcal{g}, {sf x},,{sf y},in,frak {g/h})}}. ] in a way, the curvature function measures how far a cartan geometry is from its corresponding klein geometry. in particular, a cartan geometry is locally equivalent to its corresponding klein geometry if and only if its curvature function vanishes identically ({em see} cite{sharpe}). section{theoretical background} in this thesis, we aim to effectively build the cartan geometry of real hypersurfaces in $bbb c^2$. after cartan himself in 1932, several other mathematicians reconstructed and developed this geometry, especially, chern-moser cite{chernmoser} and tanaka cite{tanaka}, who presented some alternative methods which enable one to construct the cartan geometries in higher dimension. the powerful methods of tanaka have been used widely in the important class of so-called {sl parabolic geometries}, which are a specific, rich type of cartan geometries; {em see} the recent extensive monograph cite{cap,cap-slovak} by v{c}ap and slovak. recently, ezhov, mclaughlin and schmalz published the article cite{ems} in the {em notices of the american mathematical society}, the purpose of which is to reconstruct cartans eight-dimensional coframe within tanakas framework. in this excellent expository paper, which in fact inspired us to prepare the current work, they computed again the cartan curvatures of the mentioned real hypersurfaces in $bbb c^2$, taking account of the corresponding lie algebra second cohomology space. contrary to what is sometimes believed, neither cartans computations (cite{ cartan}), nor cherns computations (cite{ chernmoser, jacobowitz, isaev}) are really effective, though, {em potentially}, they should be so after some (hard) work. ezhov, mclaughlin and schmalz (cite{ ems}) made a normalization of an initial frame for $tm$ which requires an application of the cauchy-kowalewski theorem, hence requires real analyticity of $m$ ({em cf.}~cite{ jacobowitz, merkerporten} for some {sc pde} aspects of cr geometry). by performing an alternative choice ${ h_1, h_2, t}$ of an initial frame for $tm$ which is explicit in terms of a local graphing function $varphi ( x, y, u)$ for $m$, we deviate from the normalization made in~cite{ ems} (with a more geometric-minded approach), our computational objective being to provide a cartan-tanaka connection all elements of which are completely effective in terms of $varphi (x, y, u)$,,---,,assuming only ${cal c}^6$-smoothness of $m$. one important obstacle on the way to performing completely explicit computations is that one has to divide by a complicated levi-form factor $upsilon$ ({em see} below) and then to execute several further differentiations of algebraic expressions involving $upsilon$, {em see} the functions $a_i$, $a_{ i, k_1}$, $a_{ i, k_1, k_2}$, $a_{ i, k_1, k_2, k_3}$ below whose full expansion costs hundreds of lines to maple. section{results and discussion} let $m^3subsetbbb c^2$ be a local levi-nondegenerate real 3-dimensional hypersurface passing through the origin, represented in coordinates $(z,w)=(x+iy,u+iv)$ as a graph: [ v = varphi(x,y,u) = x^2+y^2+{ m o}(3), ] for a certain real-valued ${cal c}^6$-smooth graphing function $varphi$ defined in a neighborhood of the origin in $bbb r^3$. throughout the paper, ${cal c}^6$-smoothness will be regularly assumed, because all objects (curvatures, frames, coframes) will happen to depend upon partial derivatives of order $leqslant 6$ of $varphi$. in this thesis, our intention is to reformulate cartans results in terms of the graphing function $varphi$, which is the initial, single datum of this study. the goal is to build the cartan geometry of such hypersurfaces $m^3$ and to characterize explicitly when they are locally biholomorphic to the distinguished {sl heisenberg sphere} $bbb h^3$ defined by the simplest equation having no ${ m o} ( 3)$ remainder: [ v = x^2+y^2 { m or equivalently:} w-overline{w} = 2i,zoverline{z}. ] to do this, we use tanakas powerful methods in several steps. at first, we compute the lie algebra $frak{hol}(bbb h^3)$ of infinitesimal cr automorphisms of the heisenberg sphere, namely the $bbb r$-linear space of all $(1,0)$-vector fields: [ {sf x}=z(z,w)frac{partial}{partial z}+w(z,w)frac{partial}{partial w} ] having holomorphic coefficients $z$ and $w$, whose real part is tangent to $bbb h^3$, {em i.e.}: [ ig({sf x}+overline {sf x}ig)|_{bbb h^3}equiv 0. ] easy computations yield at first some standard, known generators: egin{proposition} label{8-dim} the lie algebra $frak{hol}(bbb h^3)$ of infinitesimal cr automorphisms of the heisenberg sphere $bbb h^3$ in $bbb c^2$ is of dimension $8$ and is generated by the following eight $bbb r$-linearly independent fields: egin{eqnarray*} & {sf h}_1& := partial_z+2iz,partial_w {sf h}_2 := i, partial_z+2z,partial_w, {sf i}_1 := (w+2iz^2),partial_z+2izw,partial_w, & {sf i}_2 &:= (iw+2z^2),partial_z+2zw, {sf t} := partial_w, {sf d} := z,partial_z+2w,partial_w, & {sf j} &:= zw,partial_z+w^2,partial_w, {sf r} := iz,partial_z. end{eqnarray*} end{proposition} this lie algebra is two-graded of the form: [ frak{hol}(bbb h^3)=underbrace{frak l_{-2}oplusfrak l_{-1}}_{frak l_-}oplusfrak l_{0}oplusfrak l_{1}oplusfrak l_2, ] where: $frak {l}_{-2} = bbb{ r} , {sf t}$; $frak {l}_{ -1} = bbb{r}, {sf h}_1 oplus bbb{r}, {sf h}_2$; $frak { l}_0 = bbb{r}, {sf d} oplus bbb{r}, {sf r}$; $frak { l}_1 = bbb{r}, {sf i}_2 oplus bbb{r}, {sf i}_2$; $frak {l}_{ 2} = bbb{r} , {sf j}$. it is known ({em see} cite{beloshapka2}) that $frak l_-$ is in fact the levi-tanaka symbol algebra of every levi nondegenerate real hypersurface $m^3subsetbbb c^2$, up to isomorphism. as for the second step, we apply the tanaka prolongation procedure to the nilpotent lie algebra $frak l_-$ just above, which we rename $frak { g}_- = frak { g}_{ -2} oplus frak { g}_{ -1}$, and we recover an eight-dimensional two-graded algebra: [ frak g=underbrace{frak g_{-2}oplusfrak g_{-1}}_{frak g_-}oplusfrak g_{0}oplusfrak g_{1}oplusfrak g_2 ] which is isomorphic to $frak{hol} (bbb h^3)$ {em via} a trivial map: [ {sf t} ightarrow{sf t}, {sf h}_1 ightarrow{sf h}_1, {sf h}_2 ightarrow{sf h}_2, {sf d} ightarrow{sf d}, {sf r} ightarrow{sf r}, {sf i}_1 ightarrow{sf i}_1, {sf i}_2 ightarrow{sf i}_2, {sf j} ightarrow{sf j}. ] where ${sf t},{sf h}_1,{sf h}_2,{sf r},{sf d},{sf i}_1,{sf i}_2$, ${sf j}$ are the eight generators of $frak g$ we construct. knowing that, generally speaking, the lie algebra of infinitesimal automorphisms of a non-holonomic homogeneous distribution is isomorphic to the tanaka prolongation of its nilpotent $frak h_-$-part (cite{tanaka, yamaguchi}), our computations verify this fact in the specific case of levi-nondegenerate real hypersurfaces $m^3subsetbbb c^2$ ({em cf.} also~cite{ems}) . afterward, we compute the second cohomology of the obtained pair of graded tanaka-type lie algebras $(frak { g}_-, frak { g})$. this enables us to find in advance some significant algebraic properties of the desired curvature function $kappa$ before starting the main computations in order to construct the sought $frak g$-valued connection. for example, we can find the homogeneity of the first nonzero homogeneous component of this curvature function, and also, we can find in advance how many essential curvature components there are. to compute the cohomology space, we have used the implementation of an algorithm provided in cite{aams} which is workable within the maple software. the last section of this thesis is devoted to introduce this algorithm which is prepared in the more general case of {sl lie (super) algebras}. next, we start the computation of an initial frame for any hypersurface $m^3$ in $bbb c^2$. at first we construct two basis elements $h_1$ and $h_2$ for the complex tangent bundle $t^cm$ in terms of the defining function $varphi$ and we get: egin{lemma} for any local ${cal c}^6$-smooth hypersurface $m^3$ of $bbb c^2$ which is represented as a graph: [ v = varphi(x,y,u) ] in coordinates $(z, w) = ( x + iy, , u + iv)$, the complex tangent bundle $t^cm = { m re}, t^{ 0, 1}m$ is generated by the two explicit vector fields: egin{eqnarray*} left{ egin{array}{c} h_1 := frac{partial}{partial x} + igg( frac{varphi_y-varphi_x,varphi_u}{1+varphi_u^2} igg) frac{partial}{partial u}, h_2 := frac{partial}{partial y} + igg( frac{-varphi_x-varphi_y,varphi_u}{1+varphi_u^2} igg) frac{partial}{partial u}. end{array} ight. end{eqnarray*} end{lemma} oindent next, assuming that $m$ is furthermore levi nondegenerate, we also compute the lie bracket $t:=frac{1}{4}[h_1,h_2]$ in terms of the defining function. for each hypersurface $m^3$ defined as the graph of the function $varphi$, the associated vector fields $h_1,h_2$ and $t$ constitute a local frame on $m$, which will be what we call the {sl initial frame}. for later use, we also compute the two length-three brackets $[h_1,t]$ and $[h_2,t]$ and, fortunately, we see that both of them are certain multiples of $t$: egin{lemma} label{h-t-brackets} allowing the two notational coincidences: $x_1equiv x$ and $x_2equiv y$, one has: [ [h_1,t]=phi_1,t { m and} [h_2,t]=phi_2,t, ] where the two rational functions $phi_1$ and $phi_2$ of the variables $(x_1,x_2,u)$ are of the form: [ phi_1=frac{a_1}{delta^2upsilon} { m and} phi_2=frac{a_2}{delta^2upsilon}, ] in which the two functions $delta$ and $upsilon$ (the levi-form factor, nonzero by assumption) have the explicit expressions: egin{eqnarray*} delta & = & 1+varphi_u^2, upsilon & = & -varphi_{xx}-varphi_{yy} - 2,varphi_y,varphi_{xu} - varphi_x^2,varphi_{uu} + 2,varphi_x,varphi_{yu} - varphi_y^2,varphi_{uu} + && + 2,varphi_y,varphi_u,varphi_{yu} + 2,varphi_x,varphi_u,varphi_{xu} - varphi_u^2,varphi_{xx} - varphi_u^2,varphi_{yy}. end{eqnarray*} and in which the two numerators are given by: [ a_i := delta^2,upsilon_{x_i} + delta ig( -2,delta_{x_i},upsilon + lambda_i,upsilon_u - upsilon,lambda_{i,u} ig) - lambda_i,upsilon,delta_u {scriptstyle{(i,=,1,,2)}}, ] where we set: [ lambda_1 := varphi_y - varphi_x,varphi_u, lambda_2 := -,varphi_x - varphi_y,varphi_u. ] end{lemma} from now on, we are using a different normalization than ezhov, mclaughlin and schmalz (cite{ ems}), so that the computations begin to be substantially distinct. one should notice here that the two functions $phi_1$ and $phi_2$ which encode the lie structure of the initial frame $ig{ h_1, h_2, t ig}$ already necessitate a division by the complicated function $upsilon$, which coincides, in the real coordinates $(x, y, u)$, with the levi determinant of $m$, of course of size $1 imes 1$, because ${ m crdim}(m) = 1$. then the $a_i$ require differentiations of $upsilon$, and furthermore, higher order invariants of the normal tanaka connection we will construct,,---,,which corresponds to a known, basic parabolic geometry,,---,,will require further partial differentiations of $upsilon$ up to order $4$. this will make computations really explode when expressing back everything in terms of partial derivatives of the graphing function $varphi ( x, y, u)$ of order $leqslant 6$. in particular, we shall have to introduce furthermore the $h_k$-iterated derivatives of the functions $phi_i$ up to order $3$, where $i, k_1, k_2, k_3 = 1, 2$: [ h_{k_1}(phi_i) = { extstyle{frac{a_{i,k_1}}{delta^4,upsilon^2}}}, h_{k_2}(h_{k_1}(phi_i)) = frac{a_{i,k_1,k_2}}{delta^6,upsilon^3}, h_{k_3}(h_{k_2}(h_{k_1}(phi_i))) = { extstyle{frac{a_{i,k_1,k_2,k_3}}{delta^8,upsilon^4}}}. ] smallskip egin{proposition} label{aikkk} all the numerators appearing above are explicitly given by: egin{eqnarray*} scriptsize a_i & := &delta^2,upsilon_{x_i} + delta ig( -2,delta_{x_i},upsilon + lambda_i,upsilon_u - upsilon,lambda_{i,u} ig) - lambda_i,upsilon,delta_u, a_{i,k_1} & :=& delta^2 ig( upsilon,a_{i,x_{k_1}} - upsilon_{x_{k_1}},a_i ig) + delta ig( -2,delta_{x_{k_1}},upsilon,a_i + upsilon,lambda_{k_1},a_{i,u} - upsilon_u,lambda_{k_1},a_i ig) - 2,delta_u,upsilon,lambda_{k_1},a_i, a_{i,k_1,k_2} & := & delta^2 ig( upsilon,a_{i,k_1,x_{k_2}} - 2,upsilon_{x_{k_2}},a_{i,k_1} ig) + delta ig( -4,delta_{x_{k_2}},upsilon,a_{i,k_1} + upsilon,lambda_{k_2},a_{i,k_1,u} - 2,upsilon_u,lambda_{k_2},a_{i,k_1} ig) - & & - 4,delta_u,upsilon,lambda_{k_2},a_{i,k_1}, a_{i,k_1,k_2,k_3} & := &delta^2 ig( upsilon,a_{i,k_1,k_2,x_{k_3}} - 3,upsilon_{x_{k_3}},a_{i,k_1,k_2} ig) + delta ig( -6,delta_{x_{k_3}},upsilon,a_{i,k_1,k_2} + upsilon,lambda_{k_3},a_{i,k_1,k_2,u} - && -3,upsilon_u,lambda_{k_3},a_{i,k_1,k_2} ig) - 6,delta_u,upsilon,lambda_{k_3},a_{i,k_1,k_2}. end{eqnarray*} furthermore, these iterated derivatives identically satisfy: [ h_2(phi_1) equiv h_1(phi_2) ] and also: egin{eqnarray*} scriptsize 0 & equiv & -,h_1(h_2(h_1(phi_2))) + 2,h_2(h_1(h_1(phi_2))) - h_2(h_2(h_1(phi_1))) - phi_2,h_1(h_2(phi_1)) + phi_2,h_2(h_1(phi_1)), & equiv & -,h_2(h_1(h_1(phi_2))) + 2,h_1(h_2(h_1(phi_2))) - h_1(h_1(h_2(phi_2))) - phi_1,h_2(h_1(phi_2)) + phi_1,h_1(h_2(phi_2)), & equiv & -,h_1(h_1(h_1(phi_2))) + 2,h_1(h_2(h_1(phi_1))) - h_2(h_1(h_1(phi_1))) + phi_1,h_1(h_1(phi_2)) - phi_1,h_2(h_1(phi_1)), & equiv & -,h_2(h_2(h_1(phi_2))) + 2,h_2(h_1(h_2(phi_2))) - h_1(h_2(h_2(phi_2))) + phi_2,h_2(h_1(phi_2)) - phi_2,h_1(h_2(phi_2)). end{eqnarray*} end{proposition} the latter statement corresponds to an observation which seems to be new in the subject, seemingly absent in the papers~cite{cartan, chernmoser, jacobowitz, nurowskisparling, le, ems, isaev}. section{conclusions} subsequently, we will be able to start the main computations of the curvature function $kappa$. we compute in fact {sl curvature coefficients} $kappa^{p_{j_1}p_{j_2}}_{q_j}$, which, by definition, are the coefficients of the basis elements: [ {sf p}_{j_1}^astwedge{sf p}_{j_2}^astotimes{sf q}_j,,,,,,, {scriptstyle{({sf p}_{j_1},,{sf p}_{j_2},in,frak g_-, ,,,, {sf q}_j,in,frak g)}} ] of the vector space ${ m linig(lambda^2frak g_-,frak gig)}$, in the expression of $kappa$. here is our main result. egin{theorem} associated to any ${cal c}^6$-smooth levi-nondegenerate real $3$-dimensional hypersurface $m^3 subset bbb{ c}^2$, represented in coordinate $(z,w):=(x+iy,u+iv)$ as a graph: [ v=varphi(x,y,u)=x^2+y^2+{ m o}(3), ] there is a unique $frak { g}$-valued cartan connection which is normal and regular in the sense of tanaka. its curvature function reduces to: egin{eqnarray*} kappa(p) & = & kappa^{h_1t}_{i_1}(p), {sf h}_1^astwedge{sf t}^ast otimes {sf i}_1 + kappa^{h_1t}_{i_2}(p), {sf h}_1^astwedge{sf t}^ast otimes {sf i}_2 + kappa^{h_2t}_{i_1}(p), {sf h}_2^astwedge{sf t}^ast otimes {sf i}_1 + && + kappa^{h_2t}_{i_2}(p), {sf h}_2^astwedge{sf t}^ast otimes {sf i}_2 + kappa^{h_1t}_j(p), {sf h}_1^astwedge{sf t}^ast otimes {sf j} + kappa^{h_2t}_j(p), {sf h}_2^astwedge{sf t}^ast otimes {sf j}, end{eqnarray*} where the two main curvature coefficients, having homogeneity $4$, are of the form: egin{eqnarray*} footnotesize kappa_{i_1}^{h_1t}(p) & = & -,mathbf{delta_1},c^4 - 2,mathbf{delta_4},c^3d - 2,mathbf{delta_4},cd^3 + mathbf{delta_1},d^4, kappa_{i_2}^{h_1t}(p) & = & -,mathbf{delta_4},c^4 + 2,mathbf{delta_1},c^3d + 2,mathbf{delta_1},cd^3 + mathbf{delta_4},d^4, end{eqnarray*} in which the two functions $mathbf{ delta_1}$ and $mathbf{ delta_4}$ of only the three variables $(x, y, u)$ are {em explicitly} given by: egin{eqnarray*} scriptsize mathbf{delta_1} & =& { extstyle{frac{1}{384}}} ig[ h_1(h_1(h_1(phi_1))) - h_2(h_2(h_2(phi_2))) + 11,h_1(h_2(h_1(phi_2))) - 11,h_2(h_1(h_2(phi_1))) + && + 6,phi_2,h_2(h_1(phi_1)) - 6,phi_1,h_1(h_2(phi_2)) - 3,phi_2,h_1(h_1(phi_2)) + 3,phi_1,h_2(h_2(phi_1)) - & & - 3,phi_1,h_1(h_1(phi_1)) + 3,phi_2,h_2(h_2(phi_2)) - ig[h_1(phi_1)ig]^2 + ig[h_2(phi_2)ig]^2 - && -,2,(phi_2)^2,h_1(phi_1) + 2,(phi_1)^2,h_2(phi_2) - 2,(phi_2)^2,h_2(phi_2) + 2,(phi_1)^2,h_1(phi_1) ig], mathbf{delta_4} & = & { extstyle{frac{1}{384}}} ig[ -,3,h_2(h_1(h_2(phi_2))) - 3,h_1(h_2(h_1(phi_1))) + 5,h_1(h_2(h_2(phi_2))) + 5,h_2(h_1(h_1(phi_1))) + && +4,phi_1,h_1(h_1(phi_2)) + 4,phi_2,h_2(h_1(phi_2)) - 3,phi_2,h_1(h_1(phi_1)) - 3,phi_1,h_2(h_2(phi_2)) - && -,7,phi_2,h_1(h_2(phi_2)) - 7,phi_1,h_2(h_1(phi_1)) - 2,h_1(phi_1),h_1(phi_2) - 2,h_2(phi_2),h_2(phi_1) + && +4,phi_1phi_2,h_1(phi_1) + 4,phi_1phi_2,h_2(phi_2) ig], end{eqnarray*} and where the remaining four secondary curvature coefficients are given by: egin{eqnarray*} kappa_{i_1}^{h_2t} & =& kappa_{i_2}^{h_1t}, kappa_{i_2}^{h_2t} & =& -,kappa_{i_1}^{h_1t}, kappa^{h_1t}_j & =& widehat{h}_1ig(kappa^{h_2t}_{i_2}ig) - widehat{h}_2ig(kappa^{h_1t}_{i_2}ig), kappa^{h_2t}_j & = &-widehat{h}_1ig(kappa^{h_2t}_{i_1}ig) + widehat{h}_2ig(kappa^{h_1t}_{i_1}ig). end{eqnarray*} end{theorem} egin{corollary} a ${cal c}^6$-smooth levi nondegenerate local hypersurface $m^3 subset bbb{ c}^2$ is biholomorphic to $bbb{ h}^3$, namely is {sl spherical}, if and only if $0 equiv mathbf{ delta_1} equiv mathbf{ delta_4}$, identically as functions of $(x, y, u)$. end{corollary} the proof is just an application of the frobenius theorem (cite{ sharpe}), real analyticity of $m$ being forced by these two zero curvature equations.