Page 344 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 27: Principal Curvatures

example of a flat space-time. There are other examples of flat geometries in both settings, Notes
though. A torus or a cylinder can both be given flat metrics, but differ in their topology. Other
topologies are also possible for curved space.

27.4 Generalizations

Figure 27.1

In above figure, Parallel transporting a vector from A  N  B yields a different vector. This failure to
return to the initial vector is measured by the holonomy of the surface.
The mathematical notion of curvature is also defined in much more general contexts. Many of
these generalizations emphasize different aspects of the curvature as it is understood in lower
dimensions.
One such generalization is kinematic. The curvature of a curve can naturally be considered as a
kinematic quantity, representing the force felt by a certain observer moving along the curve;
analogously, curvature in higher dimensions can be regarded as a kind of tidal force (this is one
way of thinking of the sectional curvature). This generalization of curvature depends on how
nearby test particles diverge or converge when they are allowed to move freely in the space.
Another broad generalization of curvature comes from the study of parallel transport on a
surface. For instance, if a vector is moved around a loop on the surface of a sphere keeping
parallel throughout the motion, then the final position of the vector may not be the same as the
initial position of the vector. This phenomenon is known as holonomy. Various generalizations
capture in an abstract form this idea of curvature as a measure of holonomy. A closely related
notion of curvature comes from gauge theory in physics, where the curvature represents a field
and a vector potential for the field is a quantity that is in general path-dependent: it may change
if an observer moves around a loop.
Two more generalizations of curvature are the scalar curvature and Ricci curvature. In a curved
surface such as the sphere, the area of a disc on the surface differs from the area of a disc of the
same radius in flat space. This difference (in a suitable limit) is measured by the scalar curvature.
The difference in area of a sector of the disc is measured by the Ricci curvature. Each of the scalar
curvature and Ricci curvature are defined in analogous ways in three and higher dimensions.
They are particularly important in relativity theory, where they both appear on the side of
Einstein’s field equations that represents the geometry of spacetime (the other side of which
represents the presence of matter and energy). These generalizations of curvature underlie, for
instance, the notion that curvature can be a property of a measure.

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