Exploring Curved Spaces and Geodesics in Manifolds

1. Curved spaces Picture a smooth surface: Spaces like this, where there are no kinks, are called manifolds. They can have any dimension.

2. Curved spaces Our main requirement is that at any point, a small enough region looks like a plane: Point, P Plane, tangent at P (Or like Rnif we’re in n dimensions) Let’s look very closely near P.

3. Curved spaces Let’s look very closely near P. Our main requirement is that at any point, a small enough region looks like a plane: P A very small patch of the tangent plane is almost indistinguishable from the surface. Slightly curved surface

4. Curved spaces A side view may help. Tangent plane at P P Surface near P x If we plot the surface relative to the plane it moves away only quadratically y = 0 + 0x - ax2

5. Geodesics A geodesic is the shortest curve lying in the surface between any two points. By adding together infinitesimal bits we can construct geodesics. In a small enough patch of the surface this curve looks straight. From outside the surface, the geodesic looks curved, but it is the straightest of straight lines infinitesimally

6. Geodesics On our smooth surface, draw a smooth curve. We want to make this curve as straight as possible while keeping it on the surface.

7. Geodesics Look at one of those tiny, planar neighborhoods. P P Require the “curve” to be a straight line through this small plane

8. Geodesics Draw the straight line through P for an infinitesimal distance, to a point Q. Q P Q P Surface

9. Geodesics Now find a small plane tangent at Q and repeat. Extend the line through Q, straight to R. R Q P R Q P Surface

10. Geodesics As long as we carry out the process infinitesimally, we get a smooth curve across the surface. This “straightest possible curve” is a geodesic

11. Length of a curve Gauss showed that two dimensional surfaces, such as the surface of a sphere, have properties that can be described using only 2-dimensional information. For example, the length of an infinitesimal curve, ds, may be found from the Pythagorean theorem: R dq ds R sinq df ds2= R2dq2+ R2sin2q df2

12. Length of a curve The length of any curve and the area of any shape can be found by combining infinitesimal bits. Integrating, we can find the length of any curve. R dq ds s R sinq df ds s = R (dq2+ R2sin2q df2)1/2

13. Gauss showed that we can also characterize the curvature of any two dimensional surface at each point using only the infinitesimal Pythagorean theorem. For instance, we can find the radius of a sphere just by knowing the distances between pairs of points on the surface. R

14. At any point of any smooth 2-dimensional, we can find the radii of the maximum and minimum best-fit circles. Rmin Rmax 1 The inverse product is the curvature, C = RmaxRmin