In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.

The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of the key terms in the Einstein field equations. Furthermore, this scalar curvature is the Lagrangian density for the Einstein–Hilbert action, the Euler–Lagrange equations of which are the Einstein field equations in vacuum.

The geometry of Riemannian metrics with positive scalar curvature has been widely studied. On noncompact spaces, this is the context of the positive mass theorem proved by Richard Schoen and Shing-Tung Yau in the 1970s, and reproved soon after by Edward Witten with different techniques. Schoen and Yau, and independently Mikhael Gromov and Blaine Lawson, developed a number of fundamental results on the topology of closed manifolds supporting metrics of positive scalar curvature. In combination with their results, Grigori Perelman's construction of Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case.

Given a Riemannian metric g, the scalar curvature Scal is defined as the trace of the Ricci curvature tensor with respect to the metric:

The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace. In terms of local coordinates one can write, using the Einstein notation convention, that:

where R_ij = Ric(∂_i, ∂_j) are the components of the Ricci tensor in the coordinate basis, and where g^ij are the inverse metric components, i.e. the components of the inverse of the matrix of metric components g_ij = g(∂_i, ∂_j). Based upon the Ricci curvature being a sum of sectional curvatures, it is possible to also express the scalar curvature as

where Sec denotes the sectional curvature and e₁, ..., e_n is any orthonormal frame at p. By similar reasoning, the scalar curvature is twice the trace of the curvature operator. Alternatively, given the coordinate-based definition of Ricci curvature in terms of the Christoffel symbols, it is possible to express scalar curvature as

where ${\displaystyle {\Gamma ^{\mu }}_{\nu \lambda }}$ are the Christoffel symbols of the metric, and ${\displaystyle {\Gamma ^{\mu }}_{\nu \lambda ,\sigma }}$ is the partial derivative of ${\displaystyle {\Gamma ^{\mu }}_{\nu \lambda }}$ in the σ-coordinate direction.