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The investigation of Ricci solitons is the focus of this work. We have provedtriviality results for compact gradient Ricci soliton under certainrestriction. Later, a rigidity result is derived for a compact gradientshrinking Ricci soliton. Also, we have estimated the conjugate radius fornon-compact gradient shrinking Ricci solitons with superharmonic potential.Moreover, an upper bound for the conjugate radius of Ricci soliton withconcircular potential vector field is determined. Finally, it is proved that anon-compact gradient Ricci soliton with a pole and non-negative Ricci curvatureis non-shrinking.

Triviality Results and Conjugate Radius Estimation of Ricci Solitons