The Riemann Curvature Vector in IR3 Curvature of the Torus

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MT 318 Diﬀerential Geometry Universidad de Las Americas - Puebla April 30, 2010

The Riemann Curvature Vector in IR3

The Riemann curvature bivector is speciﬁed by

R(a ∧ b) = ∂v ∧ PaPb(v).

In the case of a 2-surface in IR3, since a ∧ b = Ia × b, we write

−1 −1 I R(Ia × b) = I ∂v ∧ PaPb(v) = ∂v × PaPb(v).

We can now express the quantity

−1 R(a ∧ b) · (c ∧ d) =< R(a ∧ b)I I(c × d) >0= [∂v × PaPb(v)] · (c × d). (1)

1 2 In terms of the basis {x1, x2} and reciprocal basis {x , x }, the classical com- i ponents R jkl are given by i i ∧ · ∧ i × · × R jkl = (x xj) R(xk xl) = (x xj) [∂v PkPl(v)]. We call ∑2 m Rij = ∂v × PiPj(v) = x × PiPj(xm) (2) m=1 the Riemann curvature vector of the 2-dimensional surface x(u, v) in IR3. The Riemann curvature vector is a scalar multiple of the unit normal vector n to the surface. The contraction R(b) = ∂a · R(a ∧ b) of the Riemann curvature bivector is called the Ricci tensor. We calculate

R(b) = ∂a · R(a ∧ b) = PaPb(∂a) − PbPa(∂a) 1 2 − 1 − 2 = Px1 Pb(x ) + Px2 Pb(x ) PbPx1 (x ) PbPx2 (x ). (3)

Curvature of the Torus

We now ﬁnd the various kinds of curvature for a torus. The torus can be parameterized by

x(u, v) = (x, y, z) = ( (c + a cos v) cos u, (c + a cos v) sin u, a sin v ) ,

where√ both u, v ∈ [0, 2π]. Each point x(u, v) on the torus satisfies the equation (c − x2 + y2)2 + z2 = a2. The torus x(u, v) is pictured in the figure. Calculating the basis and reciprocal basis of the tangent space, we find

x1 = (−(c + a cos(v)) sin(u), cos(u)(c + a cos(v)), 0)

1 1 0.5 0 2 -0.5 -1 0 -2 0 -2 2

Figure 1: Plot of torus x(u, v) for c = 2, a = 1.

x2 = (−a cos(u) sin(v), −a sin(u) sin(v), a cos(v))

x11 = (− cos(u)(c + a cos(v)), −(c + a cos(v)) sin(u), 0)

x12 = (a sin(u) sin(v), −a cos(u) sin(v), 0)

x22 = (−a cos(u) cos(v), −a cos(v) sin(u), −a sin(v)) The metric [g] is found to be ( ) (c + a cos(v))2 0 [g] = 0 a2 with the inverse ( ) 1 −1 (c+a cos(v))2 0 [g] = 1 0 a2 from which we calculate ( ) sin(u) cos(u) x1 = − , , 0 c + a cos(v) c + a cos(v) and ( ) cos(u) sin(v) sin(u) sin(v) cos(v) x2 = − , − , . a a a The projection operator

1 2 Px(a) = a · x x1 + a · x x2, × and the unit normal to the surface is n = x1 x2 , or |x1×x2| ( ) n = cos(u) cos(v)), cos(v) sin(u), sin(v)

Since Px projects onto the tangent space, and n is orthogonal to the tangent space, it follows that

Px(n) = 0 =⇒ P˙x(n) + Px(n˙ ) = 0 =⇒ P˙x(n) = −Px(n˙ ),

2 from which follow the special cases ˙ − − − Px1 (n) = Px(n(x1)) = n(x1) = ( cos(v) sin(u), cos(u) cos(v), 0) and ˙ − − − Px2 (n) = n(x2) = ( cos(u) sin(v), sin(u) sin(v), cos(v))

We also calculate Px1 (x2) = 0, ( ) ˙ − 2 − 2 − Px1 (x1) = cos(u) cos (v)(c+a cos(v)), cos (v)(c+a cos(v)) sin(u), cos(v)(c+a cos(v)) sin(v) and ( ) ˙ − − − Px2 (x2) = a cos(u) cos(v), a cos(v) sin(u), a sin(v) .

Calculation of curvature

The matrix [n] of the linear mapping n(a) is ( ) ( ) 1 · 1 · − cos(v) x n(x1) x n(x2) c+a cos(v) 0 [n] = 2 2 = 1 x · n(x1) x · n(x2) − 0 a The Gausian curvature cos(v) K = det[n] = . G cos(v)a2 + ca The Mean curvature 1 −c − 2a cos(v) κ = trace[n] = . m 2 2a(c + a cos(v))

The principal curvatures κ1, κ2 are the eigenvalues of the matrix [n], { } 1 cos(v) {κ , κ } = − , − , 1 2 a c + a cos(v) and the corresponding eigenvectors (0, 1) and (1, 0) of [n] are the principal di- rections at the point x(u, v). Finally, using (2), we calculate the Riemann curvature vectors R11 = 0 = R22, and { } 2 2 R12 = − cos(u) cos (v), − cos (v) sin(u), − cos(v) sin(v) . Using (3), we calculate the Ricci tensors { } cos(v) sin(u) cos(u) cos(v) R(x ) = , − , 0 1 a a and { } cos(u) cos(v) sin(v) cos(v) sin(u) sin(v) cos2(v) R(x ) = , , − 2 c + a cos(v) c + a cos(v) c + a cos(v)

3 · − cos(v)(c+a cos(v)) · − a cos(v) Note that R(x1) x1 = a , and R(x2) x2 = c+a cos(v) . The Riemannian scalar curvature is the constraction ∂a · R(a) of Ricci curvature. Calculating, we ﬁnd that

2 cos(v) ∂ · R(a) = x1 · R(x ) + x2 · R(x ) = − . a 1 2 cos(v)a2 + ca

The Gauss-Bonnet Theorem

Let M be a compact 2-surface in IR3. The quantity ∫ ∫

KG|dx(2)| M is called the total curvature of M. Let D be a rectangular decomposition of M, and let v, e and f be the number of vertices, edges, and faces in D. Then the integer χ = v − e + f is called the Euler-Poincare characteristic of M. Theorem: Gauss-Bonnet Theorem. If M is a compact 2-surface M in IR3 with boundary β(M) and with the Euler characteristic χ, then ∫ ∫ ∫

KG|dx(2)| + kg|dx| = 2πχ M β(M) where KG is the Gaussian curvature and kg is the geodesic curvature∫ on the M | | boundary. If the boundary β( ) is piecewise smooth, then the integral β(M) kg dx is the sum of the integrals along the smooth portions plus the sum of angles turned at the corners of the boundary β(M). For example, the surface of a cube M has v = 8, e = 12 and f = 6, so that its Euler-Poincare characteristic is χ = v −e+f = 2. All of the curvature of a cube is concentrated at its vertices, so it is not clear how to integrate over the surface of the cube. However, if the cube is “blown-up” into a sphere, it will evenly spread out its curvature, without changing its Euler-Poincare characteristic. By the Gauss-Bonnet Theorem it follows that for the 2-sphere S, ∫ ∫

KG|dx(2)| = 2πχ = 4π. S