• Bulletin of the Korean Mathematical Society
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• v.41 no.2
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• pp.213-220
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• 2004

In this paper, we generalize the Bishop-Gromov volume comparison theorem by considering an integral bound for the part of the radial Ricci curvature which lies below a given smooth function. We also establish a compactness theorem from this result.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.613-618
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• 1996

In this paper, M will always denote an n- dimensional complete Riemannian manifold and $\omega_n$ is the volume of $S^n$.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.577-585
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• 2008

In this paper, we shall calculate the volume of normal tubes around geodesics under a curvature perturbation to establish a theorem of volume comparison type.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.163-173
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• 1995

Inspired in [2,9,10,17], pp. E. Ehrlich and S. B. Kim in [4] cultivated the Riccati equation related to the Raychaudhuri equation of General Relativity for the stable Jacobi tensor along the geodesics to extend the Hawking-Penrose conjugacy theorem to $$ f(t) = Ric(c(t)',c'(t)) + tr(\sigma(A)^2) $$ where $\sigma(A)$ is the shear tensor of A for the stable Jacobi tensor A with $A(t_0) = Id$ along the complete Riemannian or complete nonspacelike geodesics c.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.421-437
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• 2019

We establish some Hessian comparison theorems and volume comparison theorems for Riemann-Finsler manifolds under various line radial integral curvature bounds. As their applications, we obtain some results on first eigenvalue, Gromov pre-compactness and generalized Myers theorem for Riemann-Finsler manifolds under suitable line radial integral curvature bounds. Our results are new even in the Riemannian case.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.33-46
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• 2023

Geodesic orbit spaces are homogeneous Finsler spaces whose geodesics are all orbits of one-parameter subgroups of isometries. Such Finsler spaces have vanishing S-curvature and hold the Bishop-Gromov volume comparison theorem. In this paper, we obtain a complete description of invariant (α₁, α₂)-metrics on spheres with vanishing S-curvature. Also, we give a description of invariant geodesic orbit (α₁, α₂)-metrics on spheres. We mainly show that a Sp(n + 1)-invariant (α₁, α2)-metric on S⁴ⁿ⁺³ = Sp(n + 1)/Sp(n) is geodesic orbit with respect to Sp(n + 1) if and only if it is Sp(n + 1)Sp(1)-invariant. As an interesting consequence, we find infinitely many Finsler spheres with vanishing S-curvature which are not geodesic orbit spaces.