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We deal here with the conjecture of  Quantum Unique Ergodicity  in a specific case. Rudnick and Sarnak showed in [Rudnick Z. and Sarnak P.: The Behaviour of Eigenstates of Arithmetic Hyperbolic Manifolds. Communications in Mathematical Physics  161  (1994), 195–213] that there is no  strong scarring  on closed geodesics for compact arithmetic congruence surfaces derived from quaternion division algebras (see theorem 1.1). We extend this theorem to a class ( K  2   S  ) of Riemannian manifoldsderived from quaternion division algebras, and show that there is no strong scarring on closed geodesics or on Γ  R  -closed imbedded totally geodesic surfaces in these manifolds (see theorem 4.1).

On the scarring of eigenstates in some arithmetic hyperbolic manifolds