By C. E. Weatherburn

ISBN-10: 0521091888

ISBN-13: 9780521091886

The aim of this ebook is to bridge the distance among differential geometry of Euclidean area of 3 dimensions and the extra complex paintings on differential geometry of generalised house. the topic is handled due to the Tensor Calculus, that is linked to the names of Ricci and Levi-Civita; and the ebook 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 through Mayer in 1930, and effectively utilized through different mathematicians.

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**Additional resources for An Introduction to Riemannian Geometry**

**Sample text**

2. Let O(m) be the orthogonal group. (i) Find a basis for the tangent space Te O(m), (ii) construct a non-vanishing vector ﬁeld Z ∈ C ∞ (T O(m)), (iii) determine all smooth vector ﬁelds on O(2). The Hairy Ball Theorem. If m ∈ N+ then there does not exist a non-vanishing vector ﬁeld X ∈ C 0 (T S 2m ). 3. Let m ∈ N+ . Use the Hairy Ball Theorem to prove that the tangent bundle T S 2m of S 2m is not trivial. Construct a nonvanishing vector ﬁeld X ∈ C ∞ (T S 2m+1 ). 4. 13. 5. Let {∂/∂xk | k = 1, . .

Xr )p ) → B(X1 , . . , Xr )(p). 2. Let M be a smooth manifold. A Riemannian metric on M is a tensor ﬁeld g : C2∞ (T M) → C0∞ (T M) such that for each p ∈ M the restriction gp = g|Tp M ⊗Tp M : Tp M ⊗ Tp M → R with gp : (Xp , Yp ) → g(X, Y )(p) is an inner product on the tangent space Tp M. The pair (M, g) is called a Riemannian manifold. The study of Riemannian manifolds is called Riemannian Geometry. Geometric properties of (M, g) which only depend on the metric g are called intrinsic or metric properties.

The best known such examples are the Sasaki and Cheeger-Gromoll metrics. For a detailed survey on the geometry of tangent bundles equipped with these metrics we recommend the paper S. Gudmundsson, E. Kappos, On the geometry of tangent bundles, Expo. Math. 20 (2002), 1-41. 52 5. 1. For m ∈ N+ let the stereographic projection 4 , Rm ) πm : (S m − {(1, 0, . . , 0)}, , Rm+1 ) → (Rm , (1 + |x|2 )2 be given by 1 πm : (x0 , . . , xm ) → (x1 , . . , xm ). 1 − x0 Prove that πm is an isometry for each m.

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