Complex Riemannian metric (holomorphic metric)

When studying open subsets $U\subseteq {\mathbb{C}}^2$ equipped with a $\mathbb{C}$-bilinear form, you enter the domain of holomorphic Riemannian geometry (or complex Riemannian geometry).

Unlike standard Kähler or Hermitian geometry — which uses a sesquilinear form (see Hermitian form, inner product and symplectic form relationship) to define a positive-definite real metric — a $\mathbb{C}$-bilinear form is complex-linear in both arguments and complex-symmetric. This creates a geometric framework that mirrors classical Riemannian geometry algebraically, but exhibits distinct features due to the algebraic closure of $\mathbb{C}$.

Formal Definition & Local Representation

Let $U\subseteq {\mathbb{C}}^2$ be an open subset with local complex coordinates $z=(z_1,z_2)$.

A holomorphic metric (or complex Riemannian metric) on $U$ is a holomorphic symmetric bilinear tensor $g\in \mathrm{\Gamma }(U,{\mathrm{Sym}}^2(T^{\ast }U))$. At each point $z\in U$, $g_z$ is a mapping

In the standard coordinate basis $\{\partial /\partial z_1,\partial /\partial z_2\}$, $g$ is represented as a matrix of holomorphic functions

The associated line element (quadratic form) is

If $det(g(z))=g_{11}{g}_{22}-g_{12}^2\ne 0$ for all $z\in U$, $g$ is non-degenerate, making $(U,g)$ a 2-dimensional complex Riemannian manifold.

Key Algebraic Peculiarities

Because the underlying field is $\mathbb{C}$ rather than $\mathbb{R}$, two critical properties distinguish complex bilinear metrics from real Riemannian metrics.

No Metric Signature

Over $\mathbb{R}$, symmetric bilinear forms are classified by rank and signature $(p,q)$ (e.g., Euclidean $(+,+)$ vs. Lorentzian $(+,-)$). Over $\mathbb{C}$, all non-degenerate symmetric bilinear forms are algebraically equivalent.

By a local change of basis, any constant non-degenerate matrix can be diagonalized to the identity matrix $I_2$. Thus there is no notion of "signature" in complex surface geometry.

Isotropic (Null) Vectors

Because $\mathbb{C}$ is algebraically closed, there always exist non-zero tangent vectors $v\ne 0$ such that

These are called isotropic or null vectors.

Example: In the flat model metric $d{s}^2=d{z}_1^2+d{z}_2^2$, the vector $v=(1,i{)}^T$ is null because $1^2+i^2=0$.

At every point $z\in U$, the equation $g_{11}{v}_1^2+2g_{12}{v}_1{v}_2+g_{22}{v}_2^2=0$ defines two complex null lines passing through the origin of $T_{z}U$, forming the null cone (or lightcone).

Differential Geometry: Connections & Curvature

The fundamental machinery of Riemannian geometry translates directly into the complex setting.

Complex Levi-Civita Connection

By the Fundamental Theorem of Complex Riemannian Geometry, there exists a unique torsion-free affine connection $\mathrm{\nabla }$ on $TU$ that preserves $g$ ($\mathrm{\nabla }g=0$).

The local Christoffel symbols are computed using the standard Koszul formula

where $g^{kl}$ denotes the entries of the inverse matrix $g^{-1}$.

Complex Geodesics

A holomorphic curve $\gamma (t)$ ($t\in \mathbb{C}$) is a geodesic if it satisfies the complex differential equation

• If $g(\dot{\gamma },\dot{\gamma })\ne 0$, the curve is a regular geodesic.
• If $g(\dot{\gamma },\dot{\gamma })=0$, the curve is a null geodesic.

Curvature and Flatness

The Riemann curvature tensor $R_{jkl}^i$, Ricci tensor $R_{ij}$, and scalar curvature $R(z)$ are defined in the usual way and yield holomorphic functions on $U$.

In complex dimension 2:

• The curvature tensor is completely specified by a single scalar holomorphic function $R(z)\in \mathbb{C}$.
• Flatness Theorem: If $R(z)\equiv 0$ on $U$, then $U$ is locally isometric to standard complex Euclidean space $({\mathbb{C}}^2,d{z}_1^2+d{z}_2^2)$.

Complex Bilinear vs. Hermitian Metrics

It is vital to contrast a $\mathbb{C}$-bilinear form with the standard Hermitian metric on ${\mathbb{C}}^2$: