Geodesics without conjugate points and curvatures at infinity.

Area of Science:

• Differential Geometry
• Geometric Analysis

Background:

• Understanding the behavior of curvature is fundamental in differential geometry.
• Theorems by Myers and Cohn-Vossen provide crucial insights into curvature properties on manifolds.

Purpose of the Study:

• To analyze the asymptotic behavior of curvature.
• To establish bounds on the integral of curvature along specific types of geodesics.
• To generalize established theorems concerning curvature.

Main Methods:

• Investigation of geodesic paths in manifolds.
• Asymptotic analysis of curvature functions.
• Application of integral calculus to geometric quantities.

Main Results:

• The integral of curvature along geodesics without conjugate points is nonpositive.
• Generalizations of Myers theorem and Cohn-Vossen's theorem are established.
• The study provides specific applications of these findings.

Conclusions:

• The nonpositivity of integrated curvature offers new constraints on geometric structures.
• The generalized theorems provide broader applicability in geometric analysis.
• The results contribute to a deeper understanding of curvature's role in shaping manifolds.