In this paper we consider the performance analysis of a digital communication system under the hypothesis that the signal constellations are on Riemannian manifolds. As a consequence of this new approach, it was necessary to extend the concepts related to signal constellations, symbol error probability and average energy of the signal constellation to the theory of differentiable manifolds. The important result coming out of this formalism is that the sectional curvature of the variety is a relevant parameter in the design and in the performance analysis of signal constellations. © Springer-Verlag Berlin Heidelberg 2003.2643 191203Agustini, E., (2002) Signal Constellations in Hyperbolic Spaces, , PhD Dissertation, IMECC-UNICAMP, Brazil, (in Portuguese)Da Silva, E.B., (2000) Signal Constellations and Its Performance Analysis in Hyperbolic Spaces, , PhD Dissertation, FEEC-UNICAMP, Brazil, (in Portuguese)Do Carmo, M.P., (1976) Differential Geometry of Curves and Surfaces, , Prentice-Hall Inc., New JerseyDo Carmo, M.P., (1979) Geometria Riemanniana, , Impa-Projeto EuclidesForney Jr., G.D., Geometrically uniform codes (1991) IEEE Trans. Inform. Theory, IT-37, pp. 1241-1260. , SeptDe Lima, J.D., Palazzo Jr., R., Embedding Discrete Memoryless Channels on Compact and Minimal Surfaces (2002) IEEE Information Theory Workshop, , Bangalore, India, Oct. 20-25Proakis, J.G., (1989) Digital Communications, 2nd Edition, , McGraw-HillCavalcante, R.G., (2002) Performance Analysis of Signal Constellations in Riemannian Varieties, , MS Thesis, FEEC-UNICAMP, Brazil, (in Portuguese)Klingenberg, W., (1978) A Course in Differential Geometry, , Springer, New York