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Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ J_{\delta , \beta}(X^v)=-\beta X^v - \delta X^h, \end{equation} where $X^h$ and $X^v$ are the horizontal and vertical lifts of $X$ and $\alpha , \delta , \beta :TS^n(1) ‎\longrightarrow \mathbb{R}$ are smooth functions satisfying in $\alpha \delta - \beta ^2 =1$(We suppose that $\beta$ is not a constant function, i.e., $d\beta \neq 0$). Also, we suppose $(x^1,...,x^n,y^1,..,y^n)=(x,y)$ is a local chart for $TS^n(1)$.

Let $z=-\frac{1-\sqrt{-1}\beta}{\sqrt{-1}\delta}$. We know that $J_{\delta , \beta}$ is a complex structure if and only if the following n complex equations hold:

\begin{equation} \frac{\partial z}{\partial x^i}+ z\frac{\partial z}{\partial y^i}=0, \hspace{1cm} i=1,...,n, \end{equation} Is there any solution for this equations?

Specially, we need such functions $u$ and $v$ which are not in the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$ where $E(x,y)=\lambda ^2(x)\sum _{i=1}^n (y^i)^2$.

In this coordinates the metric g is of the form $g=\lambda ^2 \sum _{i=1}^n dx^i ‎\otimes dx^i$. Moreover, $y^1,...,y^n$ are directly given by $x^1,...,x^n$, for example if $v=\sum _{i=1}^ny^i\frac{\partial}{\partial x^i}(x)$ then it's coordinates are $(x^1,...,x^n,y^1,...,y^n)$. Also, we want to know local solutions.

I know what the locally solutions are when $u,v$ are functions of the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$. I want to know is there any solution except these solutions?

I have some questions on Robert Bryant's answer:

1. Are our leaves of foliation $‎\lbrace(u,v)‎\rbrace \times H_+$?
2. What is the proof of the proposition in the answer?
3. Why the functions $\delta, \beta$ of the associated complex structure $J_{\delta, \beta}$ (as the associated complex structure for the equation $Z.Z+1=0$) are not functions of $E(x,y)=\lambda^2 \sum_{i=1}^n (y^i)^2$? And are there other types of solutions?

Let $(S^n,g)$ denote the unit $n$-sphere endowed with its induced metric $g$ from its embedding into $\mathbb{R}^{n+1}$. The Levi-Civita connection of $g$ induces a splitting of the tangent bundle of $\pi:TS^n\to S^n$ into horizontal and vertical parts as $TTS^n = V\oplus H$, where each summand is canonically isomorphic to $\pi^*(TS^n)$. Thus, every vector field $X$ on $S^n$ can be uniquely lifted to $TS^n$ as either a horizontal vector field $X^h$ or a vertical vector field $X^v$.

Given a triple of (smooth) real-valued functions $(\alpha,\delta,\beta)$ on $TS^n$ satisfying $\alpha \delta - \beta ^2 = 1$, one can define an almost-complex structure $J_{\delta, \beta}$ on $TS^n$ by the conditions \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ J_{\delta , \beta}(X^v)=-\beta X^v - \delta X^h, \end{equation} where $X^h$ and $X^v$ are the horizontal and vertical lifts of any vector field $X$ on $S^n$.

Question: What are the possibilities for $(\alpha,\beta,\delta)$ if we require that $J_{\delta,\beta}$ be integrable?

Remarks:

(1) I am particularly interested in integrable $J_{\delta,\beta}$ for which $\beta$ is not constant. I am also interested in understanding the integrable $J_{\delta,\beta}$ in which $(\alpha,\beta,\delta)$ are only defined on some open subset of $TS^n$.

(2) I already know some local solutions that take a special form with respect to conformal coordinates on $(S^n,g)$: If $x = (x_1,\ldots,x_n)$ are local conformal coordinates on $U\subset S^n$, so that $g = \lambda^2(x) \sum _{i=1}^n dx_i ‎\otimes dx_i$, let $y_i:TU\to\mathbb{R}$ defined by $y_i(v) = \mathrm{d}x_i(v)$ be the associated tangential coordinates. Then one can compute that a basis for the $(1,0)$-forms for $J_{\delta,\beta}$ on $TU$ are given by $$ \zeta_k = \mathrm{d}y_k + y_j(\delta_{jk}\,\mu_l\,\mathrm{d}x_l +\mu_j\,\mathrm{d}x_k-\mu_k\,\mathrm{d}x_j) - z\, \mathrm{d}x_k $$ where the summation convention is assumed and $$ \mu_j = \frac{\partial (\log\lambda)}{\partial x_j}\qquad\text{and}\qquad z = \frac{i+\beta}{\delta}. $$ Then $J_{\delta,\beta}$ is integrable if and only if $$ \mathrm{d}\zeta_k\equiv0\mod \zeta_1,\ldots,\zeta_n\,. $$ When $n>1$, this works out to be $2n$ first-order partial differential equations for $z$, so the system is overdetermined (and nonlinear). I already know the solutions when one supposes that the function $z$ is assumed to be a function of $E(x,y)=\lambda ^2(x)\sum _{i=1}^n (y_i)^2$. I want to know whether there are any solutions that are not of this form.

Addendum: I have some questions on Robert Bryant's answer:

1. Are our leaves of foliation $‎\lbrace(u,v)‎\rbrace \times H_+$?
2. What is the proof of the proposition in the answer?
3. Why the functions $\delta, \beta$ of the associated complex structure $J_{\delta, \beta}$ (as the associated complex structure for the equation $Z.Z+1=0$) are not functions of $E(x,y)=\lambda^2 \sum_{i=1}^n (y^i)^2$? And are there other types of solutions?

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ J_{\delta , \beta}(X^v)=-\beta X^v - \delta X^h, \end{equation} where $X^h$ and $X^v$ are the horizontal and vertical lifts of $X$ and $\alpha , \delta , \beta :TS^n(1) ‎\longrightarrow \mathbb{R}$ are smooth functions satisfying in $\alpha \delta - \beta ^2 =1$(We suppose that $\beta$ is not a constant function, i.e., $d\beta \neq 0$). Also, we suppose $(x^1,...,x^n,y^1,..,y^n)=(x,y)$ is a local chart for $TS^n(1)$.

we know that $J_{\delta , \beta}$ is a complex structure if and only if the following equations hold:

\begin{equation} \frac{\partial u}{\partial y^i}=0,\hspace{1cm} i=1,...,n,\\ \lambda ^2(x) y^i +\frac{\partial u}{\partial x^i}=v\frac{\partial v}{\partial y^i},\hspace{1cm} i=1,...,n,\\ \frac{\partial v}{\partial x^i}=-u\frac{\partial v}{\partial y^i},\hspace{1cm} i=1,...,n, \end{equation} where $u=\frac{\beta}{\delta}$ and $v=\frac{1}{\delta}$. Is there any solution for this equations?

Specially, we need such functions $u$ and $v$ which are not in the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$ where $E(x,y)=\lambda ^2(x)\sum _{i=1}^n (y^i)^2$.

In this coordinates the metric g is of the form $g=\lambda ^2 \sum _{i=1}^n dx^i ‎\otimes dx^i$. Moreover, $y^1,...,y^n$ are directly given by $x^1,...,x^n$, for example if $v=\sum _{i=1}^ny^i\frac{\partial}{\partial x^i}(x)$ then it's coordinates are $(x^1,...,x^n,y^1,...,y^n)$. Also, we want to know local solutions.

I know what the locally solutions are when $u,v$ are functions of the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$. I want to know is there any solution except these solutions?

I have some questions on Robert Bryant's answer:

1. Are our leaves of foliation $‎\lbrace(u,v)‎\rbrace \times H_+$?
2. What is the proof of the proposition in the answer?
3. Why the functions $\delta, \beta$ of the associated complex structure $J_{\delta, \beta}$ (as the associated complex structure for the equation $Z.Z+1=0$) are not functions of $E(x,y)=\lambda^2 \sum_{i=1}^n (y^i)^2$? And are there other types of solutions?

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ J_{\delta , \beta}(X^v)=-\beta X^v - \delta X^h, \end{equation} where $X^h$ and $X^v$ are the horizontal and vertical lifts of $X$ and $\alpha , \delta , \beta :TS^n(1) ‎\longrightarrow \mathbb{R}$ are smooth functions satisfying in $\alpha \delta - \beta ^2 =1$(We suppose that $\beta$ is not a constant function, i.e., $d\beta \neq 0$). Also, we suppose $(x^1,...,x^n,y^1,..,y^n)=(x,y)$ is a local chart for $TS^n(1)$.

Let $z=-\frac{1-\sqrt{-1}\beta}{\sqrt{-1}\delta}$. We know that $J_{\delta , \beta}$ is a complex structure if and only if the following n complex equations hold:

\begin{equation} \frac{\partial z}{\partial x^i}+ z\frac{\partial z}{\partial y^i}=0, \hspace{1cm} i=1,...,n, \end{equation} Is there any solution for this equations?

Specially, we need such functions $u$ and $v$ which are not in the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$ where $E(x,y)=\lambda ^2(x)\sum _{i=1}^n (y^i)^2$.

In this coordinates the metric g is of the form $g=\lambda ^2 \sum _{i=1}^n dx^i ‎\otimes dx^i$. Moreover, $y^1,...,y^n$ are directly given by $x^1,...,x^n$, for example if $v=\sum _{i=1}^ny^i\frac{\partial}{\partial x^i}(x)$ then it's coordinates are $(x^1,...,x^n,y^1,...,y^n)$. Also, we want to know local solutions.

I know what the locally solutions are when $u,v$ are functions of the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$. I want to know is there any solution except these solutions?

I have some questions on Robert Bryant's answer:

1. Are our leaves of foliation $‎\lbrace(u,v)‎\rbrace \times H_+$?
2. What is the proof of the proposition in the answer?
3. Why the functions $\delta, \beta$ of the associated complex structure $J_{\delta, \beta}$ (as the associated complex structure for the equation $Z.Z+1=0$) are not functions of $E(x,y)=\lambda^2 \sum_{i=1}^n (y^i)^2$? And are there other types of solutions?

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ J_{\delta , \beta}(X^v)=-\beta X^v - \delta X^h, \end{equation} where $X^h$ and $X^v$ are the horizontal and vertical lifts of $X$ and $\alpha , \delta , \beta :TS^n(1) ‎\longrightarrow \mathbb{R}$ are smooth functions satisfying in $\alpha \delta - \beta ^2 =1$(We suppose that $\beta$ is not a constant function, i.e., $d\beta \neq 0$). Also, we suppose $(x^1,...,x^n,y^1,..,y^n)=(x,y)$ is a local chart for $TS^n(1)$.

we know that $J_{\delta , \beta}$ is a complex structure if and only if the following equations hold:

\begin{equation} \frac{\partial u}{\partial y^i}=0,\hspace{1cm} i=1,...,n,\\ \lambda ^2(x) y^i +\frac{\partial u}{\partial x^i}=v\frac{\partial v}{\partial y^i},\hspace{1cm} i=1,...,n,\\ \frac{\partial v}{\partial x^i}=-u\frac{\partial v}{\partial y^i},\hspace{1cm} i=1,...,n, \end{equation} where $u=\frac{\beta}{\delta}$ and $v=\frac{1}{\delta}$. Is there any solution for this equations?

Specially, we need such functions $u$ and $v$ which are not in the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$ where $E(x,y)=\lambda ^2(x)\sum _{i=1}^n (y^i)^2$.

In this coordinates the metric g is of the form $g=\lambda ^2 \sum _{i=1}^n dx^i ‎\otimes dx^i$. Moreover, $y^1,...,y^n$ are directly given by $x^1,...,x^n$, for example if $v=\sum _{i=1}^ny^i\frac{\partial}{\partial x^i}(x)$ then it's coordinates are $(x^1,...,x^n,y^1,...,y^n)$. Also, we want to know local solutions.

I know what the locally solutions are when $u,v$ are functions of the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$. I want to know is there any solution except these solutions?

I have some questions on Robert Bryant's answer:

1. Are our leaves of foliation ${(u,v)} \times H_+$?
2. What is the proof of the proposition in the answer?
3. Why the functions $\delta, \beta$ of the associated complex structure $J_{\delta, \beta}$ (as the associated complex structure for the equation $Z.Z+1=0$) are not functions of $E(x,y)=\lambda^2 \sum_{i=1}^n (y^i)^2$? And are the other type of solutions?

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition \begin{equation} J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\ J_{\delta , \beta}(X^v)=-\beta X^v - \delta X^h, \end{equation} where $X^h$ and $X^v$ are the horizontal and vertical lifts of $X$ and $\alpha , \delta , \beta :TS^n(1) ‎\longrightarrow \mathbb{R}$ are smooth functions satisfying in $\alpha \delta - \beta ^2 =1$(We suppose that $\beta$ is not a constant function, i.e., $d\beta \neq 0$). Also, we suppose $(x^1,...,x^n,y^1,..,y^n)=(x,y)$ is a local chart for $TS^n(1)$.

we know that $J_{\delta , \beta}$ is a complex structure if and only if the following equations hold:

\begin{equation} \frac{\partial u}{\partial y^i}=0,\hspace{1cm} i=1,...,n,\\ \lambda ^2(x) y^i +\frac{\partial u}{\partial x^i}=v\frac{\partial v}{\partial y^i},\hspace{1cm} i=1,...,n,\\ \frac{\partial v}{\partial x^i}=-u\frac{\partial v}{\partial y^i},\hspace{1cm} i=1,...,n, \end{equation} where $u=\frac{\beta}{\delta}$ and $v=\frac{1}{\delta}$. Is there any solution for this equations?

Specially, we need such functions $u$ and $v$ which are not in the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$ where $E(x,y)=\lambda ^2(x)\sum _{i=1}^n (y^i)^2$.

In this coordinates the metric g is of the form $g=\lambda ^2 \sum _{i=1}^n dx^i ‎\otimes dx^i$. Moreover, $y^1,...,y^n$ are directly given by $x^1,...,x^n$, for example if $v=\sum _{i=1}^ny^i\frac{\partial}{\partial x^i}(x)$ then it's coordinates are $(x^1,...,x^n,y^1,...,y^n)$. Also, we want to know local solutions.

I know what the locally solutions are when $u,v$ are functions of the form $u(x,y)=f(E(x,y))$ and $v(x,y)=h(E(x,y))$. I want to know is there any solution except these solutions?

I have some questions on Robert Bryant's answer:

1. Are our leaves of foliation $‎\lbrace(u,v)‎\rbrace \times H_+$?
2. What is the proof of the proposition in the answer?
3. Why the functions $\delta, \beta$ of the associated complex structure $J_{\delta, \beta}$ (as the associated complex structure for the equation $Z.Z+1=0$) are not functions of $E(x,y)=\lambda^2 \sum_{i=1}^n (y^i)^2$? And are there other types of solutions?