A statistical hypothesis test for long range dependence (LRD) in manifold-supported functional time series is formulated in the spectral domain. The proposed test statistic is based on the weighted periodogram operator, assuming that the elements of the spectral density operator family are invariant with respect to the group of isometries of the manifold. A Central Limit Theorem is derived to obtain the asymptotic Gaussian distribution of the proposed test statistics operator under the null hypothesis. The rate of convergence to zero, in the Hilbert--Schmidt operator norm, of the bias of the integrated empirical second and fourth order cumulant spectral density operators is established under the alternative hypothesis. The consistency of the test is then derived, from the obtained consistency of the integrated weighted periodogram operator under LRD. Practical implementation of our testing approach is based on the random projection methodology. The frequency-varying Karhunen-Lo\'eve expansion of invariant Gaussian random spectral Hilbert-Schmidt kernels on manifolds is considered for generation of random directions in the implementation of this methodology. A simulation study illustrates the main results regarding asymptotic normality and consistency, and the empirical size and power properties of the proposed testing approach.

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