A remark on positively curved manifolds of dimensions 7 and 13

TL;DR

This paper constructs infinitely many closed 7-manifolds as totally geodesic submanifolds within 13-dimensional positively curved closed Riemannian manifolds and discusses related geometric embedding problems.

Contribution

It introduces new constructions of totally geodesic embeddings of 7-manifolds into 13-dimensional positively curved manifolds and explores related geometric questions.

Findings

Existence of infinitely many such embeddings

Discussion on pinching constants

Analysis of other potential embeddings

Abstract

Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.