Scalar Curvature - Direct Geometric Interpretation

Direct Geometric Interpretation

When the scalar curvature is positive at a point, the volume of a small ball about the point has smaller volume than a ball of the same radius in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume of a small ball is instead larger than it would be in Euclidean space.

This can be made more quantitative, in order to characterize the precise value of the scalar curvature S at a point p of a Riemannian n-manifold . Namely, the ratio of the n-dimensional volume of a ball of radius ε in the manifold to that of a corresponding ball in Euclidean space is given, for small ε, by

\frac{\operatorname{Vol} (B_\varepsilon(p) \subset M)}{\operatorname{Vol} (B_\varepsilon(0)\subset {\mathbb R}^n)}= 1- \frac{S}{6(n+2)}\varepsilon^2 + O(\varepsilon^4).

Thus, the second derivative of this ratio, evaluated at radius ε = 0, is exactly minus the scalar curvature divided by 3(n + 2).

Boundaries of these balls are (n-1) dimensional spheres with radii ; their hypersurface measures ("areas") satisfy the following equation:

\frac{\operatorname{Area} (\partial B_\varepsilon(p) \subset M)}{\operatorname{Area} (\partial B_\varepsilon(0)\subset {\mathbb R}^n)}= 1- \frac{S}{6n}\varepsilon^2 + O(\varepsilon^4).

Read more about this topic:  Scalar Curvature

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