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Choose a normal ball on which ${\exp }_p$ is a diffeomorphism, and take $\epsilon$ smaller than half its radius. For $|X|<\epsilon$, $t\mapsto {\exp }_p(tX)$ is a geodesic of length $|X|$. If a competing curve remains in the normal ball, write it in polar form. The Gauss lemma makes radial and angular velocities orthogonal, so its speed is at least the absolute radial speed and its length is at least $|X|$. A curve leaving the larger normal ball already accumulates more than $|X|$ in radial variation. Thus the radial geodesic minimizes length.

At $q$, both $X$ and the geodesic sphere $\Sigma$ are hypersurfaces. If their tangent hyperplanes were distinct, they would be transverse. The transverse intersection theorem would then make $X\cap \Sigma$ a submanifold of dimensionFor $m\ge 3$ this has positive dimension near $q$, contradicting that the intersection is the singleton $\{q\}$. Therefore $T_{q}X=T_q\Sigma$.

The conclusion fails in dimension two because a transverse intersection is zero-dimensional and may be isolated. In the Euclidean plane, let $\Sigma$ be the unit circle, $q=(1,0)$, and let $X=\{(x,0):1/2<x<3/2\}$. Then $X\cap \Sigma =\{q\}$, but $T_{q}X$ is horizontal whereas $T_q\Sigma$ is vertical.

Solved by gpt-5.6-sol high.

The restriction of $J$ to the $J$-invariant bundle $TY$ is an almost complex structure. Its Nijenhuis tensor is the restriction of the ambient Nijenhuis tensor because vector fields tangent to an embedded submanifold have tangent Lie bracket. The ambient tensor vanishes since $X$ is a complex manifold, so the Newlander-Nirenberg theorem makes the induced structure on $Y$ integrable. The inclusion has complex-linear differential and is therefore holomorphic; hence $Y$ is a complex submanifold.

For a complex submanifold, the holomorphic normal bundle isIf $Y$ is a smooth hypersurface, taking top exterior powers in the holomorphic conormal sequencegivesThe normal bundle of a hypersurface is $N_{Y/X}\cong {\mathcal{O}}_X(Y){|}_Y$, so the Adjunction formula is

On $X={\mathbb{CP}}^n\times {\mathbb{CP}}^m$, a bihomogeneous polynomial of bidegree $(d_1,d_2)$ is a section of the holomorphic line bundle $\mathcal{O}(d_1,d_2)$. Its zero locus is smooth precisely when the section is transverse to the zero section, equivalently when $F$ and all of its homogeneous first partial derivatives have no common projective zero. Sincethe Adjunction formula yieldsThus a smooth $F$ with $d_1=n+1$ and $d_2=m+1$ defines a complex submanifold with trivial canonical bundle.

Solved by gpt-5.6-sol high.