Riemannian Geometry SCM Submanifold Levi-Civita Connection 2nd Fundamental Form [ N(X,Y) = N(Y,X) ] Thm: Let X, Y, Z, T be v.f. on S. [ \langle R^{M}(X,Y)Z,T \rangle = ] [ \langle R^{S}(X,Y)Z,T \rangle + \langle N(X,Z),N(Y,T) \rangle - \langle N(X,T),N(Y,Z) \rangle ] Claim: X, Y vector field on S [ (\nabla_{X}^{M}Y){P} = (\nabla{X}^{M}Y){P}^{\perp} + (\nabla{X}^{M}Y){P}^{\top} ] [ (\nabla{X}^{M}Y){P}^{\top} = (\nabla{X}^{S}Y){P} ] [ (\nabla{X}^{M}Y)_{P}^{\perp} = N(X,Y) ] Recall: Riemannian curvature R is a (1,3)-form valued in End(TM). Claim: R^{M} = 0 is a flat metric Note: 0 <= R^{S} <= 1/13.77 + 1/12, and non-negative curvature is a line bundle. Gauss-Bonnet Thm: Classical statement: (S, g) smooth compact surface with Riemannian metric g. Scalar Curvature: R_{ij} = R^{k}{ijk} R = R{ij}g^{ij} Ricci Tensor: R_{ij} = R^{k}_{ijk}g^{kj} Scalar curvature: R = R_{ij}g^{ij} Ex: (HW Probl.) Q: Parallel transport along a circle r on a sphere. a) S = S^2 unit sphere. Scalar curvature R = 1. Volume (S^2) = 4π. Gauss-Bonnet: ∫_S R dvol = 2πχ(S) 4π = 2πχ(S) χ(S) = 2 b) S = T^2 with flat metric. LHS: ∫_S R dvol = 0 RHS: 2πχ(T^2) = 2π(2-2) = 0 dvol_n = dx_1 ∧ ... ∧ dx_n dvol_n = (det(A)) dx_1 ∧ ... ∧ dx_n G_ij = g(∂/∂x_i, ∂/∂x_j) G_ij = g((Σ(∂y_a/∂x_i)∂/∂y_a), (Σ(∂y_b/∂x_j)∂/∂y_b)) = Σ(∂y_a/∂x_i)(∂y_b/∂x_j)δ_ab = A^T A det(G) = det(A^T A) = det(A)^2 |e_1 x ... x e_n| = √det(g(e_i, e_j))