An Introduction to Riemannian Geometry by C. E. Weatherburn

By C. E. Weatherburn

The aim of this publication is to bridge the distance among differential geometry of Euclidean house of 3 dimensions and the extra complex paintings on differential geometry of generalised area. the topic is handled through the Tensor Calculus, that's linked to the names of Ricci and Levi-Civita; and the publication offers an creation either to this calculus and to Riemannian geometry. The geometry of subspaces has been significantly simplified by means of use of the generalized covariant differentiation brought by way of Mayer in 1930, and effectively utilized by means of different mathematicians.

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R. Then B(X1 , . . , Xr )(p) = B(Y1 , . . , Yr )(p). Proof. We shall prove the statement for r = 1 the rest follows by induction. Put X = X1 and Y = Y1 and let (U, x) be local coordinates on M. Choose a function f ∈ C ∞ (M) such that f (p) = 1 and 43 44 5. RIEMANNIAN MANIFOLDS support(f ) is contained in U. Then define v1 , . . , vm ∈ C ∞ (T M) by (vk )q = f (q) · ( ∂x∂ k )q 0 if q ∈ U if q ∈ /U Then there exist functions ρk , σk ∈ C ∞ (M) such that m f ·X = m and f · Y = ρk vk k=1 σk vk . k=1 Now m B(X)(p) = f (p)B(X)(p) = B(f · X)(p) = ρk (p)B(vk )(p) k=1 and similarily m σk (p)B(vk )(p).

Then it is easily seen that the following equations hold g(∇XY , Z) = X(g(Y, Z)) − g(Y, ∇XZ), g(∇XY , Z) = g([X, Y ], Z) + g(∇Y X, Z) = g([X, Y ], Z) + Y (g(X, Z)) − g(X, ∇Y Z), 0 = −Z(g(X, Y )) + g(∇ZX, Y ) + g(X, ∇ZY ) = −Z(g(X, Y )) + g(∇XZ + [Z, X], Y ) + g(X, ∇Y Z − [Y, Z]). 6. THE LEVI-CIVITA CONNECTION 55 By adding these relations we yield 2 · g(∇XY , Z) = {X(g(Y, Z)) + Y (g(Z, X)) − Z(g(X, Y )) +g(Z, [X, Y ]) + g(Y, [Z, X]) − g(X, [Y, Z])}. If {E1 , . . , Em } is a local orthonormal frame for the tangent bundle then m ∇XY = k=1 g(∇XY , Ei )Ei .

By C ∞ (E) we denote the set of all smooth vector fields of (E, M, π). From now on we shall, when not stating otherwise, assume that all our vector bundles are smooth. 4. Let (E, M, π) be a vector bundle over a manifold M. Then we define the operations + and · on the set C ∞ (E) of smooth sections of (E, M, π) by (i) (v + w)p = vp + wp , (ii) (f · v)p = f (p) · vp for all v, w ∈ C ∞ (E) and f ∈ C ∞ (M). If U is an open subset of M then a set {v1 , . . , vn } of vector fields v1 , . . , vn : U → E on U is called a local frame for E if for each p ∈ U the set {(v1 )p , .

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