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A smooth mapping ~ of a Riemannian manifold MI onto a Riemannian manifold M2 is said to be conformal if there exists a positive function % = %~ on MI such that for every x~M I the mapping ~(x).d~x is an isometry of the tangent spaces (MI) x and (M2)~x ~. A conformal diffeomorphism ~ is called a homothety if %~ = const.Two Riemannian manifolds MI and M2 are said to be conformally equivalent (homothetic) if there exists a conformml diffeomorphism (homothety) of M~ onto M2.If M~ is conformally equivalent to M2, then we will write MI ~ M2. A Riemannian manifold is said to be conformally flat if each of its points has a neighborhood which is conformally equivalent to an open ball in a Euclidean space.In this note we will obtain a classification of all conformally flat Riemannian manifolds, admitting a transitive group of conformal transformations.We denote the complete group of conformal transformations (motions) by CM (I(M)), and its connected component of the identity by Co(M) (Io(M)).The following Riemannian manifolds are conformally flat:E n --the n-dimensional Euclidean space (n>i);T n --the n-dimensional (n>i) flat torus; S n --the n-dimensional (n~i) sphere whose curvature in an arbitrary two-dimensional direction is equal to one for n~2 and S I is a circle of circumference 27.pn --the n-dimensional elliptic space which is obtained from the sphere S n by identifying diametrically opposite points (n>2);L~n-1 m --the manifold obtained from the sphere S 2n-~ by embedding it in the n-dimensional complex vector space in the standard manner and identifying the points obtained from each other by multiplying by an m-th root of unity (n>2, m>3); L4n-I(A) --the manifold obtained from the sphere S 4n-~ by embedding it in the n-dimensional left vector space over the skew field of quaternions and identifying the points obtained from each other by multiplying on the left by some element of the finite subgroup A* = ~-I(A) of the.multiplicative group Sp(1) of unit quaternions (n>i, T:Sp (1)-+SO{3) is the natural covering, and A is one of the following subgroups of the group S0(3): ~p ~>2), ~-, O and J are the rotation groups of a regular p-gonal prism ~>3), tetrahedron, and isocohedron respectively; ~2 being Klein's four-group);

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