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:

- Are our leaves of foliation $\lbrace(u,v)\rbrace \times H_+$?
- What is the proof of the proposition in the answer?
- 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?

arefunctions of the type you mention (i.e., they are functions of $|v|^2$). However, for the generic hyperquadric that I wrote down, they are not. This actually follows from Remark 1, since the form for $E(x,y)$ that you wrote down is not invariant under the action of the affine group of $\mathbb{R}^{n+1}$, but you can just take an explicit example when $n=1$ and check this directly. $\endgroup$ – Robert Bryant Feb 15 '16 at 15:49notthe differential equations $(u,v)$ must satisfy in order for $J_{\delta,\beta}$ be integrable. The correct equations when $n>1$ (as I show in my answer) are $2n$ in number, but you have written down $3n$ equations, which is too many. Also, it is clearly not right for $n=1$ because, when $n=1$all$J_{\delta,\beta}$ are integrable complex structures. You need to fix this, because it misleads readers. $\endgroup$ – Robert Bryant Feb 16 '16 at 13:15