# Download Intro to Differential Geometry and General Relativity by S. Waner PDF

By S. Waner

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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers according to talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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Xn) = 0. Show that Xi is identically zero in any coordinate system. 2. Give and example of a contravariant vector field that is not covariant. Justify your claim. 3. Verify the following claim If V and W are contravariant (or covariant) vector fields on M, and if å is a real number, then V+W and åV are again contravariant (or covariant) vector fields on M. 4. 7: If Ci is covariant and Vj is contravariant, then Ck Vk is a scalar. 5. Let ˙: Sn’E1 be the scalar field defined by ˙(p1, p2, . . , pn+1) = pn+1.

Thus, we can think of covariant tangent fields as nothing more than 1forms. Proof Here is the one-to-one correspondence. Let F be the family of 1-forms on M (or U) and let C be the family of covariant vector fields on M (or U). Define ∞: C’F by ∞(Ci)(Vj) = Ck Vk . In the homework, we see that Ck Vk is indeed a scalar by checking the transformation rule: C—k V—k = ClVl. The linearity property of ∞ now follows from the distributive laws of arithmetic. We now define the inverse 30 §: F’C by (§(F))i = F(∂/∂xi).

An en ·e1 + 0 V·e2 = a1 e1 ·e2 + ... + an en ·e2 ... V·en = a1 e1 ·en + ... ei]g* *. ei], 62 as required.  For reasons that will become clear later, let us now digress to look at some partial derivatives of the fundamental matrix [g* *] in terms of ambient coordinates. ∂ ∂  ∂ys ∂ys [g ] = p  q r ∂xp qr ∂x ∂x ∂x ∂2 ys ∂ys ∂2 ys ∂ys = p q r + r p q ∂x ∂x ∂x ∂x ∂x ∂x or, using “comma notation”, gqr,p = ys,pq ys,r + ys,rp ys,q Look now at what happens to the indices q, r, and p if we permute them (they're just letters, after all) cyclically in the above formula (that is, p→q→r), we get two more formulas.