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

Famous quotes containing the words direct and/or geometric:

    “Irony, forsooth! Guard yourself, Engineer, from the sort of irony that thrives up here; guard yourself altogether from taking on their mental attitude! Where irony is not a direct and classic device of oratory, not for a moment equivocal to a healthy mind, it makes for depravity, it becomes a drawback to civilization, an unclean traffic with the forces of reaction, vice and materialism.”
    —Thomas Mann (1875–1955)

    “In mathematics he was greater
    Than Tycho Brahe, or Erra Pater:
    For he, by geometric scale,
    Could take the size of pots of ale;
    Resolve, by sines and tangents straight,
    If bread and butter wanted weight;
    And wisely tell what hour o’ th’ day
    The clock doth strike, by algebra.”
    —Samuel Butler (1612–1680)