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We present a systematic study of $f$-mode oscillations in neutron starscontaining hyperons, extending recent results obtained within the Cowlingapproximation to linearized General Relativity. Employing a relativistic meanfield model, we find that the Cowling approximation can overestimate thequadrupolar $f$-mode frequency of neutron stars by up to 30\% compared to thefrequency obtained in the linearized general relativistic formalism. Imposingcurrent astrophysical constraints, we derive updated empirical relations forgravitational wave asteroseismology. The frequency and damping time ofquadrupole $f$-mode oscillations of hyperonic stars are found to be in therange of 1.47 - 2.45kHz and 0.13 - 0.51 sec respectively. Our correlationstudies demonstrate that among the various parameters of the nucleonic andhyperonic sectors of the model, the nucleon effective mass shows the strongestcorrelation with mode characteristics and neutron star observables. Estimatesfor the detectability of $f$-modes in a transient burst of gravitational wavesfrom isolated hyperonic stars is also provided.

General relativistic treatment of $f$-mode oscillations of hyperonic stars