Curvature Gaussian Curvature • How curved is it? • Curvature of a surface R1 – The radius of an osculating – Draw two principal circle can be used to measure osculating circles at a given curvature of a line at a given point on the surface point. – Obtain two principal – Curvature = 1/(curvature radius) curvature radii, R1 and R2 Johann Carl Friedrich Gauss R2 • Curvature is in units of 1/length – Gaussian curvature is given (1777-1855) – The signs posted on the road by 1/(R1 R2), up to the saying “R=300ft” or “R=500ft” overall sign. • Which one is more curved? • K=Gaussian curvature • A straight line (zero curvature) – K is in units of 1/area has R=infinity

Flat, Spherical, Hyperbolic Measuring Curvature • Homogeneous and Isotropic space can be either flat, • !=Sum of the angles of a triangle minus " spherical, or hyperbolic. – !=K x (area of triangle) – K is the same everywhere • Flat – !=0 (flat) – Zero K – !>0 (spherical) R1= R2=Infinity – K=0 – !<0 (hyperbolic) • Spherical R1= R2=R – Positive K – K=1/R2 > 0 Osculating circles on the same • !=Change in direction of an arrow through side • Hyperbolic a closed circuit of the “vector transport” – Negative K R1= R2=R – !=K x (area enclosed by circuit) 2 – K= -1/R < 0 Osculating circles on opposite – This is neat – try it yourself! sides Distances in Curved Space Gauss’s Theorema Egregium • Gaussian curvature is an “intrinsic” one. • In flat space (Euclidean space), a distance rd between two points on a surface is given by – We don’t need to know anything about 3 dimension to measure Gaussian curvature of 2-dimensional space 2 2 2 – (Distance) = (x-int.) + (y-int.) • “Flatlanders” can measure curvature of their world (which – ds2 = dx2 + dy2 is 2 dimensional) without knowing anything about the 3rd • In curved space (non-Euclidean), Euclidean dimension. formula no longer applies: – Therefore, we can measure Gaussian curvature of 3- dimensional space without knowing anything about – (Distance)2 4th dimension. 2 2 = F(x-int.) + 2G(x-int.)(y-int.) + H(y-int.) • Theorema Egregium – ds2 = Fdx2 + 2Gdxdy + Hdy2 – How metric coefficients vary from point to point on a – F, G, H : Metric Coefficients surface contains all the information of the geometry • Metric coefficients generally depend on x and y of the surface.

Riemannian Geometry • Riemann has extended Spacetime Metric of the Universe Gauss’s work to four and higher dimensions. & dR2 # • Metric coefficients have: ds2 c2dt 2 R2 d 2 sin 2 d 2 = ' $ 2 + ( ) + ) ( )! – 3 components in 2d %1' KR " – 6 components in 3d – 10 components in 10d • Important question: What is K of the universe? – etc… – K determines curvature of 3-d space in which we are • Riemann curvature: living generalization of Gaussian • According to Gauss’s theorema egregium, we curvature in higher dimensions. can measure K, without knowing anything about the 4th dimension. • Einstein used Riemannian Georg Friedrich Bernhard Riemann geometry to construct the (1826-1866) – This implies that the shape of the observable universe general theory of relativity. can be determined!