Exersices of Diff. Geom.-II (Part of Riemnanian Geom.)
1. Consider the upper half-plane
R2+ = {(x1,x2) ∈ R2;x2 > 0}
with the Riemannian metric given by g = g = 1 ,g = 0. 11 22 (x2)2 12
1) Compute all the Christoffel symbols of the Riemannian connection Γkij; 2) Let v = (0,1) be a tangent vector at the point (0,1) of R2 (i.e. v = ∂
0 + 0 ∂x2 under the coordinate (x1,x2), which is a unit vector at the point (0,1)). Let
v(t) be the parallel transport of v0 along the curve γ(t) = (t, 1). Show that v(t) makes an angle t with the x2-curve, measured in the clockwise sense. (Hint: use the equation of parallel transport)
K(p) = Ricp(x)dS , ωn−1 S n−1
1
2. Prove that the scalar curvature K(p) at p ∈ M is given by 1 n−1
where ωn−1 is the area of the standard (n − 1)-sphere Sn−1 in TpM and dSn−1 is the volume elements on Sn−1.
3. A Riemannian manifold Mn is called an Einstein manifold if, for all X,Y ∈ X(M), Ric(X,Y) = λ < X,Y >, where λ : M → R is a real valued function. Prove that
1) If M is connected and Einstein, with n ≥ 3, then λ is constant on M;
2) If M is a connected 3-dimensional Einstein manifold then M has constant sectional curvature.
4. Let M be a Riemannian manifold. f ∈ D(M) and X ∈ X(M). The gradient of f as a vector field gradf on M is given by
< (gradf)(p),v >= dfp(v), p ∈ M, v ∈ TpM.
The divergence of X as a function divX : M → R is given by (divX)(p) = traceofthelinearmappingY(p)→(∇YX)(p),Y ∈X(M),p∈M.
Define the Laplacian operator △ : D(M) → D(M) by △f = divgradf, f ∈ D(M).
Take a local coordinate of M, (x1,x2,··· ,xn), and the Riemannian metric given g = gijdxidxj. Prove
1 ∂ √ ij∂ j △f=√g∂xi( gg ∂xf),
where g = det(gij), (gij) = (gij)−1.