I am trying to construct a Kaehler structure on $R^{4}$ following paper of V. Oproiu, "Some new geometric structures on the tangent bundle". This is how the author constructs a metric on a tangent bundle $TM$ of a Riemannian manifold $(M,g)$. We know that the Levi-Civita connection of $g$ defines a direct sum decomposition \begin{align} \notag TTM=VTM \oplus HTM, \end{align} of the tangent bundle into the vertical distribution $VTM$ and the horizontal distribution $HTM$. The vector fields $(\frac{\partial}{\partial y^{1}},..,\frac{\partial}{\partial y^{n}})$ define a local frame field for $VTM$ and for $HTM$ we have the local frame field $(\frac{\delta}{\delta x^{1}},..,\frac{\delta}{\delta x^{n}})$, where \begin{align} \notag \frac{\delta}{\delta x^{i}}=\frac{\partial}{\partial x^{i}}-\Gamma_{i0}^{h}\frac{\partial}{\partial y^{h}}, \end{align} \begin{align} \notag \Gamma_{i0}^{h}=\Gamma_{ik}^{h}y^{k}, \end{align} $\Gamma_{ik}^{h}$ are the Christoffel symbols defined by the Riemannian metric $g$.

(In my case, where $M=R^{4}$, $\frac{\delta}{\delta x^{i}}=\frac{\partial}{\partial x^{i}}$)

Next we consider the energy density \begin{align} t=\frac{1}{2}g_{ik}(x)y^{i}y^{k}. \end{align} Let $u$, $v$ be two real smooth functions. We shall assume that $u$, $u+2tv$ have positive values. Then we may consider the following symmetric $M$-tensor field of type $(0,2)$ on $TM$, defined by the components \begin{align} G_{ij}=u(t)g_{ij}+v(t)g_{0i}g_{0j}, \end{align} where $g_{0i}=g_{hi}y^{h}$. With, \begin{align} H^{kl} \end{align}

we denote the coefficients of the inverse matrix of $(G_{ij})$.

We shall also use the components $H_{ij}$ \begin{align} H_{ij}=g_{ik}H^{kl}g_{lj} \end{align}

We use the followig $M$-tensor fields on $TM$ \begin{align} G^{kl}=g^{ki}G_{ij}g^{jl}, G_{k}^{i}=G^{ih}g_{hk}, H_{k}^{i}=H^{ih}g_{hk}. \end{align} The following Riemannian manifold may be considered on $TM$ \begin{align} \notag G(\frac{\delta}{\delta x^{i}}, \frac{\delta}{\delta x^{j}})=G_{ij}, \end{align} \begin{align} \notag G(\frac{\partial}{\partial y^{i}},\frac{\partial}{\partial y^{j}})=H_{ij}, \end{align} \begin{align} \notag G(\frac{\partial}{\partial y^{i}},\frac{\delta}{\delta x^{j}})=G(\frac{\delta}{\delta x^{j}},\frac{\partial}{\partial y^{i}})=0. \end{align} An almost complex structure $J$ may be defined on $TM$ by \begin{align} J\frac{\delta}{\delta x^{i}}=G_{i}^{k}\frac{\partial}{\partial y^{k}}, \end{align} \begin{align} J\frac{\partial}{\partial y^{i}}=-H_{i}^{k}\frac{\delta}{\delta x^{k}}. \end{align} Then we have two theorems. Theorem 1

$(TM, G, J)$ is an almost Kaehlerian manifold.

Theorem 2

$J$ is integrable if $(M,g)$ has constant sectional curvature $c$ and the function $v$ is given by \begin{align} v=\frac{c-uu'}{2tu'-u}. \end{align}

Using the calculation above, I obtained this for the metric on $R^{4}$. \begin{align} \notag G = \left (\begin{array}{cccc} u+v(y^{1})^{2} & vy^{1}y^{2} & 0 & 0 \\ vy^{1}y^{2}& u+v(y^{2})^{2} & 0 & 0 \\ 0 & 0 & \alpha (u+v(y^{2})^{2})& -\alpha vy^{1}y^{2}\\ 0 & 0 & -\alpha vy^{1}y^{2} & \alpha (u+v(y^{2})^{1}) \end{array} \right ) \end{align}, where

\begin{align} \notag \alpha = \frac {1} {u^{2} + uv ((y^{1})^{2} + (y^{2})^{2})} \end{align}

and this for $J$.

\begin{align} \notag J\frac{\partial}{\partial x^{1}} = (u+v(y^{1})^{2})\frac{\partial}{\partial y^{1}} + vy^{1}y^{2}\frac{\partial}{\partial y^{2}} \end{align}

\begin{align} \notag J\frac{\partial}{\partial x^{2}} = vy^{1}y^{2}\frac{\partial}{\partial y^{1}} + (u+v(y^{2})^{2})\frac{\partial}{\partial y^{2}} \end{align}

\begin{align} \notag J\frac{\partial}{\partial y^{1}} = -\alpha (u+v(y^{2})^{2})\frac{\partial}{\partial x^{1}} + \alpha vy^{1}y^{2}\frac{\partial}{\partial x^{2}} \end{align}

\begin{align} \notag J\frac{\partial}{\partial y^{2}} = \alpha vy^{1}y^{2}\frac{\partial}{\partial x^{1}} -\alpha (u+v(y^{1})^{2})\frac{\partial}{\partial x^{2}} \end{align}

Then, by checking we obtain that $(R^{4}, G, J)$ is an almost Kaehler manifold, but not a Kaehler. Also, if I use Theorem 2 for obtaining $v$, there is a problem.

If we choose $u=(y^{1})^{2} + (y^{2})^{2}$, then according to Theorem 2, $v= -1$ and $u+2tv=0$.

Hope you can help me with this calculation and notice where the mistake is better then myself.

Thank you very much in advance.


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