In theorem 1.2 of Brian Conrad's handout Operations with Pseudo-Riemannian metrics, the author writes

Theorem 1.2. Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \infty$ admits a Riemannian metric. In particular, every smooth manifold with corners admits a structure of a Riemannian manifold with corners.

 

The main tool in the proof will be $C^p$ partitions of unity, so it is amazing that Morrey and Grauert proved that every real-analytic vector bundle over a real-analytic manifold admits a real-analytic Riemannian metric tensor. The real-analytic case rests on serious input from the theory of several complex variables in order to circumvent the lack of real-analytic partitions of unity.

What is the idea of Morrey and Grauert's proof? What's the formalism that allows circumventing partitions of unity? What are the miracles of complex analysis that make things work?

In theorem 1.2 of Brian Conrad's handout Operations with Pseudo-Riemannian metrics, the author writes

Theorem 1.2. Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \infty$ admits a Riemannian metric. In particular, every smooth manifold with corners admits a structure of a Riemannian manifold with corners.

The main tool in the proof will be $C^p$ partitions of unity, so it is amazing that Morrey and Grauert proved that every real-analytic vector bundle over a real-analytic manifold admits a real-analytic Riemannian metric tensor. The real-analytic case rests on serious input from the theory of several complex variables in order to circumvent the lack of real-analytic partitions of unity.

What is the idea of Morrey and Grauert's proof? What's the formalism that allows circumventing partitions of unity? What are the miracles of complex analysis that make things work?