Holomorphic isometric immersion

Definition

Let $(M,J_M,g_M)$ and $(N,J_N,g_N)$ be Hermitian manifolds. A holomorphic isometric immersion is a smooth map $f:M\to N$ satisfying

df\circ J_M=J_N\circ df \qquad\text{and}\qquad f^*g_N=g_M.

The first equation says that $f$ is holomorphic, while the second makes it a Riemannian isometric immersion. No separate immersion hypothesis is needed because the metric pullback equation makes $df$ injective.

Preservation of the fundamental form

Write

\omega_M(X,Y)=g_M(J_MX,Y),\qquad \omega_N(U,V)=g_N(J_NU,V).

The two defining equations imply

f^*\omega_N=\omega_M.

Hence a holomorphic isometric immersion between Kähler manifolds is also a symplectic map. Conversely, a holomorphic map between Kähler manifolds that pulls back $\omega_N$ to $\omega_M$ also pulls back $g_N$ to $g_M$ . In this setting the terms Kähler immersion and holomorphic isometric immersion are commonly synonymous.

Isomorphisms and near misses

If $f$ is a diffeomorphism, it is a biholomorphic Riemannian isometry and therefore a strict isomorphism of the Kähler data. A holomorphic map alone need not preserve either metric or Kähler form. A Riemannian isometric immersion alone need not be holomorphic.

The definition is strict: a homothetic holomorphic immersion satisfying $f^*g_N=cg_M$ for $c\neq1$ is not isometric unless the metrics are rescaled.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §3.1, Hermitian metrics and Kähler forms.
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume II, Wiley, 1969. Publisher record. Relevant: Chapter IX, Hermitian and Kähler geometry.