TY - JOUR
T1 - On the Triviality of a Certain Kind of Shrinking Solitons
AU - Zhang , Zhuhong
JO - Journal of Mathematical Study
VL - 2
SP - 169
EP - 177
PY - 2019
DA - 2019/05
SN - 52
DO - http://doi.org/10.4208/jms.v52n2.19.04
UR - https://global-sci.org/intro/article_detail/jms/13157.html
KW - Einstein manifold, shrinking gradient Ricci soliton, positive Ricci curvature, pinched sectional curvature.
AB - <p style="text-align: justify;">In this paper, we study shrinking gradient Ricci solitons whose Ricci tensor
has one eigenvalue of multiplicity at least $n−2.$ Firstly, we show that if the minimal
eigenvalue of Ricci tensor has multiplicity at least $n−1$ at each point, then the soliton
are Einstein. While on the shrinking gradient Ricci solitons whose maximal eigenvalue
has multiplicity at least $n−1,$ the triviality are also true if we naturally require the
positivity of Ricci tensor.<br/>We further prove that if the maximal (or minimal) eigenvalue of Ricci tensor has multiplicity at least $n−2$ at each point , and in addition the sectional curvature is bounded
from above, then the soliton are Einstein.</p>