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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

## Page text matches

- ==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