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We exhibit large classes of examples of noncommutative finite-dimensionalmanifolds which are (non-formal) deformations of classical manifolds. The mainresult of this paper is a complete description of noncommutativethree-dimensional spherical manifolds, a noncommutative version of the sphere$S^3$ defined by basic K-theoretic equations. We find a 3-parameter family ofdeformations $S^3_{\mathbf{u}}$ of the standard 3-sphere $S^3$ and acorresponding 3-parameter deformation of the 4-dimensional Euclidean space$\mathbb R^4$. For generic values of the deformation parameters we show thatthe obtained algebras of polynomials on the deformed $\mathbb R^4_{\mathbf{u}}$only depend on two parameters and are isomorphic to the algebras introduced bySklyanin in connection with the Yang-Baxter equation. It follows that different$S^3_{\mathbf{u}}$ can span the same $\mathbb R^4_{\mathbf{u}}$. Thisequivalence generates a foliation of the parameter space $\Sigma $. Thisfoliation admits singular leaves reduced to a point. These critical points areeither isolated or fall in two 1-parameter families $ C_{\pm}\subset \Sigma $.Up to the simple operation of taking the fixed algebra by an involution, thesetwo families are identical and we concentrate here on $ C_{+}$. For $u \inC_{+}$ the above isomorphism with the Sklyanin algebra breaks down and thecorresponding algebras are special cases of $\theta$-deformations, a notionwhich we generalize in any dimension and various contexts, and study in somedetails. Here, and this point is crucial, the dimension is not an artifact,i.e. the dimension of the classical model, but is the Hochschild dimension ofthe corresponding algebra which remains constant during the deformation....Comment: 52 pages, Section 3 rewritten, references adde

Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples