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• over \mathcal{D}^1(M). By definition, the element m(1) - 1 induces the zero map on \Gamma(E), so the map descends to a homomorphism \mathcal{C}(M) ...
4 KB (777 words) - 00:42, 24 July 2009
• ==Statement== Suppose E is a vector bundle over a differential manifold M. Denote by \mathcal{E} the sheaf of sections of E. Consider the first ...
2 KB (273 words) - 17:20, 6 January 2012
• to the de Rham derivative of a function, yielding a 1-form. ===Connection, transport along a curve=== connection along a curve, transport along a curve ...
11 KB (1,926 words) - 21:18, 24 July 2009
• {| class="sortable" border="1" ! Fact no. !! Name ... | 1 || Any connection is C^\infty-linear in its subscript argument ...
7 KB (1,442 words) - 17:36, 6 January 2012
• {| class="sortable" border="1" ! Doubly ruled surface ... \frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1 ...
766 bytes (104 words) - 14:52, 5 August 2011
• \nabla is denoted as \tau(\nabla). It is a (1,2)-tensor defined as: \tau(\nabla)(X,Y)= \nabla_XY - \nabla_YX - [X,Y]. A connection whose torsion ...
3 KB (441 words) - 17:29, 6 January 2012
• b(X,Y) = 1/2(b(X+y,X+Y) - b(X,X) - b(Y,Y)) It is also easy to see that: ... given point, then the Ricci curvature is (n-1) times that constant. Thus ...
3 KB (542 words) - 21:05, 6 January 2012
• ==Definition== Let M be a differential manifold and g be a defining ingredient::Riemannian metric or defining ingredient::pseudo-Riemannian metric ...
2 KB (266 words) - 02:28, 24 July 2009
• tensoroftype|(1,3)} ==Definition== ===As a (1,3)-tensor=== Let M be a connected differential manifold ...
4 KB (601 words) - 01:22, 24 July 2009
• {| class="sortable" border="1" ! Ruled surface ... (u,0,0) and \delta(u) is the vector (0,1,0). || The Euclidean plane ...
3 KB (483 words) - 14:50, 5 August 2011
• \sum_{1 \le i where K(e_i,e_j) denotes the sectional curvature of the ... \sum_{1 \le i \le n, 1 \le j \le n} R(e_i,e_j,e_j,e_i) ==Related notions== ...
2 KB (366 words) - 02:26, 24 July 2009
• itself be differentiated via \nabla, since R is a (1,3)-tensor and we can define the connection on all (p,q)-tensors. With this meaning, the following ...
2 KB (299 words) - 01:23, 24 July 2009
• {| class="sortable" border="1" ! Curve being revolved !! Surface of revolution |- | semicircle with endpoints for a circle ...
3 KB (483 words) - 23:30, 29 July 2011
• {| class="sortable" border="1" ! Fact no. !! Name ... | 1 || Any connection is C^\infty-linear in its subscript argument ...
4 KB (737 words) - 17:56, 6 January 2012
• (1,2) ==Definition== ===Given data=== * A differential manifold M ... The torsion map is a (1,2) tensor. It is tensorial in both X and Y ...
1 KB (226 words) - 17:57, 6 January 2012
• Consider a smooth curve \gamma:[0,1] \to M. Let D/dt denote the connection along \gamma induced by \nabla, and consider the transport along ...
1 KB (161 words) - 21:17, 6 January 2012
• Let M be a differential manifold, E a vector bundle on M. Let \gamma: ... v \mapsto \phi_t(v) (t \in [0,1]) such that for any vector v \in E_{ ...
1 KB (216 words) - 18:02, 6 January 2012
• \nabla_Y(W)) + g(\nabla_X(Z),\nabla_Y(W)) \qquad (1). And: ... Substituting (1) and (2) in (\dagger\dagger) yields (\dagger). ...
2 KB (489 words) - 01:52, 24 July 2009
• \! R(X,Y,Z,W) + R(Y,Z,X,W) + R(Z,X,Y,W) = 0 \qquad (1) Similar statements ... Consider (1) + (2) - (3) - (4) and uses facts (1) and (2). We get: ...
2 KB (373 words) - 02:24, 24 July 2009
• g(\nabla_XY,Z) + g(Y,\nabla_XZ) = Xg(Y,Z) \qquad (1) g(\nabla_YX,Z ... We now use fact (1): g(\nabla_{fX}(Y),Z) = \frac{fXg(Y,Z) + (Yf)g(Z ...
5 KB (965 words) - 18:43, 24 July 2009

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