Ivancevic_Applied-Diff-Geom

Introduction 21real number. This concept is just like a dot product, or inner product. Thisfunction from vectors into the real numbers is required to vary smoothlyfrom point to point. 26On any Riemannian manifold, from its second–order metric tensorg = 〈·, ·〉, one can derive the associated fourth–order Riemann curvaturetensor. This tensor is the most standard way to express curvature of Riemannianmanifolds, or more generally, any manifold with an affine connection,torsionless or with torsion. 2726 Once a local coordinate system x i is chosen, the metric tensor appears as a matrix,conventionally given by its components, g = g ij . Given the metric tensor of aRiemannian manifold and using the Einstein summation notation for implicit sums,the length of a segment of a curve parameterized by t, from a to b, is defined as:L = R qb dxa g i dx jij dt dt dt. Also, the angle θ between two tangent vectors ui , v i is definedgas: cos θ =ij u i v jq|gij u i u j ||g ij v i v j | .27 The Riemann curvature tensor is given in terms of a Levi–Civita connection ∇ (moregenerally, an affine connection, or covariant differentiation, see below) by the followingformula:R(u, v)w = ∇ u∇ vw − ∇ v∇ uw − ∇ [u,v] w,where u, v, w are tangent vector–fields and R(u, v) is a linear transformation of thetangent space of the manifold; it is linear in each argument. If u = ∂/∂x i and v = ∂/∂x jare coordinate vector–fields then [u, v] = 0 and therefore the above formula simplifies toR(u, v)w = ∇ u∇ vw − ∇ v∇ uw,i.e., the curvature tensor measures non–commutativity of the covariant derivative. Thelinear transformation w ↦→ R(u, v)w is also called the curvature transformation or endomorphism.In local coordinates x µ (e.g., in general relativity) the Riemann curvature tensor canbe written using the Christoffel symbols of the manifold’s Levi–Civita connection:R ρ σµν = ∂ µΓ ρ νσ − ∂νΓρ µσ + Γρ µλ Γλ νσ − Γρ νλ Γλ µσ .The Riemann curvature tensor has the following symmetries:R(u, v) = −R(v, u),R(u, v)w + R(v, w)u + R(w, u)v = 0.〈R(u, v)w, z〉 = −〈R(u, v)z, w〉,The last identity was discovered by Ricci, but is often called the first Bianchi identity oralgebraic Bianchi identity, because it looks similar to the Bianchi identity below. Thesethree identities form a complete list of symmetries of the curvature tensor, i.e. givenany tensor which satisfies the identities above, one can find a Riemannian manifold withsuch a curvature tensor at some point. Simple calculations show that such a tensor hasn 2 (n 2 − 1)/12 independent components.The Bianchi identity involves the covariant derivatives:∇ uR(v, w) + ∇ vR(w, u) + ∇ wR(u, v) = 0.