[Paper Review] A remark on positively curved manifolds of dimensions 7 and 13

I. A. Taĭmanov|ArXiv.org|Nov 30, 1995

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper constructs infinitely many closed 7-manifolds that are totally geodesically embedded into 13-dimensional positively curved Riemannian manifolds, demonstrating the existence of such embeddings in these specific dimensions. The key contribution is a geometric construction method that yields examples with controlled curvature, while also discussing pinching constants and the limitations of such 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.

Motivation & Objective

• To investigate the existence of totally geodesic embeddings of closed 7-manifolds into positively curved 13-dimensional Riemannian manifolds.
• To explore the geometric and topological constraints on such embeddings in dimensions 7 and 13.
• To analyze the pinching constants associated with the constructed manifolds.
• To examine the possibility of other totally geodesic embeddings beyond the constructed examples.

Proposed method

• The construction employs techniques from Riemannian geometry to produce explicit examples of 13-dimensional manifolds with positive sectional curvature.
• Totally geodesic embeddings of 7-manifolds are realized via symmetric space constructions and curvature estimates.
• The method relies on the existence of suitable group actions and homogeneous structures to ensure the geodesic property.
• Pinching constants are analyzed using curvature comparison theorems and metric estimates.
• The approach involves modifying known positively curved manifolds to accommodate the 7-manifold embeddings.
• The construction is verified through differential geometric arguments and consistency checks on curvature bounds.

Experimental results

Research questions

• RQ1 Can infinitely many closed 7-manifolds be totally geodesically embedded into a single 13-dimensional positively curved Riemannian manifold?
• RQ2 What are the optimal pinching constants achievable in such embeddings?
• RQ3 Are there topological obstructions to the existence of totally geodesic embeddings of 7-manifolds in 13-dimensional positively curved manifolds?
• RQ4 Can the construction be generalized to other dimensions or different types of submanifolds?
• RQ5 What is the role of symmetry and homogeneous structures in enabling such embeddings?

Key findings

• Infinitely many closed 7-manifolds are shown to admit totally geodesic embeddings into 13-dimensional positively curved Riemannian manifolds.
• The constructed 13-manifolds possess positive sectional curvature, confirming the existence of such ambient spaces.
• Pinching constants are discussed, though no explicit numerical values are computed in the paper.
• The existence of such embeddings is established through geometric and topological constructions based on symmetric spaces.
• The paper identifies limitations and open problems regarding the existence of other totally geodesic embeddings in similar settings.
• The revised version includes corrections and additions to the original construction, improving the rigor of the embedding argument.