Distributional Sectional Curvature Bounds for Riemannian Metrics of Low Regularity

Area of Science:

• Differential Geometry
• Geometric Analysis
• Topology

Background:

• Sectional curvature bounds are crucial in Riemannian geometry and Alexandrov spaces.
• Existing definitions apply to smooth or specific generalized settings.
• Need for a definition applicable to less regular metrics.

Purpose of the Study:

• Introduce a new notion of sectional curvature bounds for continuous Riemannian metrics.
• Investigate the properties of this new definition, particularly for Geroch-Traschen regularity.
• Relate the new definition to existing concepts like Alexandrov's triangle comparison.

Main Methods:

• Utilize a distributional version of the classical sectional curvature formula.
• Analyze manifolds with continuous Riemannian metrics of H^1_loc intersect C^0 regularity.
• Examine the case of locally Lipschitz continuous metrics.

Main Results:

• For C^1 metrics, the new distributional definition recovers the Alexandrov bound.
• A weaker version of this result holds for locally Lipschitz continuous metrics.
• Demonstrates the utility of distributional methods in geometric analysis.

Conclusions:

• The new distributional sectional curvature bound is a robust generalization.
• It bridges the gap between smooth and non-smooth geometric settings.
• Opens new avenues for studying spaces with lower regularity metrics.