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  • This category lists important facts about regular values, critical values, regular points and critical points of smooth maps between differential ...
    3 members (0 subcategories, 0 files) - 20:16, 18 May 2008

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  • ==Facts== ===For surfaces=== For the two-dimensional case, viz surfaces, in the special event that the surface is orientable, we can use the Gauss ...
    1 KB (199 words) - 19:33, 18 May 2008
  • ==Facts== Any Einstein metric with positive cosmological constant, for instance, satisfies the hypotheses for the Bonnet-Myers theorem, and hence ...
    3 KB (443 words) - 19:33, 18 May 2008
  • ==Facts== ===Ricci-flatness=== Yau's theorem tells us that every Calabi-Yau manifold admits a Ricci-flat metric, viz a compatible metric ...
    458 bytes (65 words) - 19:33, 18 May 2008
  • ==Facts== ===Sedykh's theorem=== Sedykh's theorem Any Caratheodory curve has at least four flattenings. ...
    371 bytes (45 words) - 19:33, 18 May 2008
  • ==Facts used== # Hopf-Rinow theorem # Nonpositively curved implies conjugate-free # Local isometry of complete Riemannian manifolds is covering map ...
    1 KB (188 words) - 13:12, 22 May 2008
  • The Chern-Weil theorem gives the basic facts about the Chern form that also establish its importance. In particular, it shows that the cohomology ...
    1 KB (190 words) - 19:34, 18 May 2008
  • Some easy facts: * Two circles centered at the same point are termed concentric circles. Given two concentric circles, there is a dilation, or ...
    3 KB (562 words) - 23:24, 29 July 2011
  • ==Facts== ===Volume-normalization of a conformal structure=== ...
    617 bytes (82 words) - 19:34, 18 May 2008
  • ==Facts== ===Connection gives connection along a curve=== Given a connection on the whole vector bundle E, we can obtain a connection along the ...
    1 KB (248 words) - 19:35, 18 May 2008
  • ==Facts== ===Viewing a connection on a vector bundle as a principal connection=== ===Transport using principal connections=== ...
    1 KB (208 words) - 19:35, 18 May 2008
  • ==Related facts== * Metric connection on metric bundle equals connection on principal O-bundle ==Definitions used== ===Connection on a vector bundle=== ...
    2 KB (319 words) - 19:36, 18 May 2008
  • ==Facts== differential 1-form|sheaf of differential 1-forms tangent bundle ...
    376 bytes (56 words) - 19:36, 18 May 2008
  • ==Related facts== * Curvature is tensorial ==Proof== The proof is based on the fact that [X,Y] = - [Y,X] We have: R(X,Y) = \nabla_X \circ \nabla_Y ...
    512 bytes (96 words) - 01:17, 24 July 2009
  • ==Related facts== * Curvature is antisymmetric in first two variables ... ==Facts used== {| class="sortable" border="1" ...
    7 KB (1,442 words) - 17:36, 6 January 2012
  • ==Facts== ===Where it lives=== The de Rham derivative can be viewed as: d:C^\infty(M) \to \Gamma(T^*M) i.e. it is a map from the algebra of infinitely ...
    2 KB (271 words) - 19:37, 18 May 2008
  • ==Facts== Every vector field on a differential manifold gives rise to a derivation, and this gives a correspondence between vector fields and ...
    738 bytes (119 words) - 19:38, 18 May 2008
  • ==Facts used== We use two main facts: # Fixed-point set of finite group of diffeomorphisms ...
    1 KB (259 words) - 19:39, 18 May 2008
  • ==Facts== ===For a planar curve=== * The length of the evolute between any two points is the total variation in the radius of curvature between ...
    602 bytes (101 words) - 19:39, 18 May 2008
  • ==Related facts== * Second Bianchi identity (also called the differential Bianchi identity). * Curvature is tensorial * Torsion is tensorial ...
    2 KB (299 words) - 01:14, 24 July 2009
  • ==Facts== ===Holonomy=== The restricted holonomy group of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has ...
    2 KB (212 words) - 19:40, 18 May 2008

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