# Canonical isometric embeddings of projective spaces into spheres

## TL;DR

This paper constructs inductive isometric embeddings of real and complex projective spaces into spheres, showing how to extend lower-dimensional embeddings to higher dimensions while preserving metrics.

## Contribution

It introduces a novel inductive method for isometric embeddings of projective spaces into spheres, aligning with the telescopic construction of infinite-dimensional projective and sphere spaces.

## Key findings

- Explicit embeddings for all dimensions of real and complex projective spaces.
- Method preserves canonical metrics through inductive extension.
- Embeddings relate real and complex projective spaces via restrictions.

## Abstract

We define inductively isometric embeddings of $\mb{P}^n(\mb{R})$ and $\mb{P}^n(\mb{C})$ (with their canonical metrics conveniently scaled) into the standard unit sphere, which present the former as the restriction of the latter to the set of real points. Our argument parallels the telescopic construction of $\mb{P}^\infty(\mb{R})$, $\mb{P}^\infty(\mb{C})$, and $\mb{S}^\infty$ in that, for each $n$, it extends the previous embedding to the attaching cell, which after a suitable renormalization makes it possible for the result to have image in the unit sphere.