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I do not like complex numbers and can make a mistake easily...

Let $L_p$ be a complex line in a tangent space $T_pX$. It is easy to see that $\exp_p$ gives an isometric embedding $L_p\hookrightarrow X$ which is also star-shaped with center at $p$; set $L=\exp_p(L_p)$

Take any other point $q\in L$, and let $L_q\subset T_q$ be the tangent subspace to $L$. Note that the maps $\exp_p$ and $\exp_q$ coinside (up to a shift) on the geodesic $(pq)$. [Here I use that if two holomorphic maps coincide on the real line then they coincide in the complex plane.] It follows that $L=\exp_q(L_q)$. Therefore $L$ is totally geodesic. In other words:

For any complex sectional direction in $X$, there is a tangent totally geodesic surface which is isometric to complex plane.

Does not it imply that $X$ is isometric to $\mathbb C^n$?

I do not like complex numbers and can make a mistake easily...

Let $L_p$ be a complex line in a tangent space $T_pX$. It is easy to see that $\exp_p$ gives an isometric embedding $L_p\hookrightarrow X$ which is also star-shaped with center at $p$; set $L=\exp_p(L_p)$

Take any other point $q\in L$, and let $L_q\subset T_q$ be the tangent subspace to $L$. Note that the maps $\exp_p$ and $\exp_q$ coinside (up to a shift) on the geodesic $(pq)$. [Here I use that if two holomorphic maps coincide on the real line then they coincide in the complex plane.] It follows that $L=\exp_q(L_q)$. Therefore $L$ is totally geodesic.

In other words: For any complex sectional direction in $X$, there is a tangent totally geodesic surface which is isometric to complex plane.

In particular, the sectional curvature in all complex sectional directions is zero and therefore the curvature of $X$ is identically zero; the later stated in Kobayashi--Nomizu, Foundations of differential geometry, Volume 2 IX, Prop. 7.1. (thanks to RdN).

I do not like complex numbers and can make a mistake easily...

Let $L_p$ be a complex line in a tangent space $T_pX$. It is easy to see that $\exp_p$ gives an isometric embedding $L_p\hookrightarrow X$ which is also star-shaped with center at $p$; set $L=\exp_p(L_p)$

Take any other point $q\in L$, and let $L_q\subset T_q$ be the tangent subspace to $L$. Note that the maps $\exp_p$ and $\exp_q$ coinside (up to a shift) on the geodesic $(pq)$. It follows that $L=\exp_q(L_q)$. Therefore $L$ is totally geodesic. In other words:

For any complex sectional direction in $X$, there is a tangent totally geodesic surface which is isometric to complex plane.

Does not it imply that $X$ is isometric to $\mathbb C^n$?

I do not like complex numbers and can make a mistake easily...

Let $L_p$ be a complex line in a tangent space $T_pX$. It is easy to see that $\exp_p$ gives an isometric embedding $L_p\hookrightarrow X$ which is also star-shaped with center at $p$; set $L=\exp_p(L_p)$

Take any other point $q\in L$, and let $L_q\subset T_q$ be the tangent subspace to $L$. Note that the maps $\exp_p$ and $\exp_q$ coinside (up to a shift) on the geodesic $(pq)$. [Here I use that if two holomorphic maps coincide on the real line then they coincide in the complex plane.] It follows that $L=\exp_q(L_q)$. Therefore $L$ is totally geodesic. In other words:

For any complex sectional direction in $X$, there is a tangent totally geodesic surface which is isometric to complex plane.

Does not it imply that $X$ is isometric to $\mathbb C^n$?

I do not like complex numbers and can make a mistake easily...

Let $L_p$ be a complex line in a tangent space $T_pX$. It is easy to see that $\exp_p$ gives an isometric embedding $L_p\hookrightarrow X$ which is also star-shaped with center at $p$; set $L=\exp_p(L_p)$

Take any other point $q\in L$, and let $L_q\subset T_q$ be the tangent subspace to $L$. Note that the maps $\exp_p$ and $\exp_q$ coinside (up to a shift) on the geodesic $(pq)$. It follows that $L=\exp_q(L_q)$. Therefore $L$ is totally geodesic.

Now $X$ contains isometric totally geodesic complex plane in every complex direction. Does not it imply that $X$ is isometric to $\mathbb C^n$?

I do not like complex numbers and can make a mistake easily...

Let $L_p$ be a complex line in a tangent space $T_pX$. It is easy to see that $\exp_p$ gives an isometric embedding $L_p\hookrightarrow X$ which is also star-shaped with center at $p$; set $L=\exp_p(L_p)$

Take any other point $q\in L$, and let $L_q\subset T_q$ be the tangent subspace to $L$. Note that the maps $\exp_p$ and $\exp_q$ coinside (up to a shift) on the geodesic $(pq)$. It follows that $L=\exp_q(L_q)$. Therefore $L$ is totally geodesic. In other words:

For any complex sectional direction in $X$, there is a tangent totally geodesic surface which is isometric to complex plane.

Does not it imply that $X$ is isometric to $\mathbb C^n$?