Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2024; 61(2): 433-450

Online first article March 13, 2024      Printed March 31, 2024

https://doi.org/10.4134/BKMS.b230136

Copyright © The Korean Mathematical Society.

Geometric inequalities for affine connections on Riemannian manifolds

Huiting Chang, Fanqi Zeng

Xinyang Normal University; Xinyang Normal University

Abstract

Using a Reilly type integral formula due to Li and Xia \cite{LiXia2017}, we prove several geometric inequalities for affine connections on Riemannian manifolds. We obtain some general De Lellis-Topping type inequalities associated with affine connections. These not only permit to derive quickly many well-known De Lellis-Topping type inequalities, but also supply a new De Lellis-Topping type inequality when the $1$-Bakry-\'{E}mery Ricci curvature is bounded from below by a negative function. On the other hand, we also achieve some Lichnerowicz type estimate for the first (nonzero) eigenvalue of the affine Laplacian with the Robin boundary condition on Riemannian manifolds.

Keywords: Reilly type formula, affine connection, De Lellis-Topping inequality, eigenvalue

MSC numbers: Primary 53C21; Secondary 58J32

Supported by: The research of authors is supported by NSFC (No.12101530), the Science and Technology Project of Henan Province (No.232102310321), and the Key Scientific Research Program in Universities of Henan Province (Nos.21A110021, 22A110021) and Nanhu Scholars Program for Young Scholars of XYNU (No.2019).