Second fundamental form: Difference between revisions

Revision as of 10:45, 31 August 2007

Definition

Let $M$ be a manifold embedded in $R^{n}$ . the second fundamental form on $M$ is the map $Γ (T M) \times Γ (T M) \to Γ (R^{n})$ defined as follows:

$S (X, Y) = \nabla_{X} Y - \nabla'_{X} Y$

where $\nabla$ si the Levi-Civita connection of $R^{n}$ and $\nabla^{'}$ is the Levi-Civita connection on $M$ .

@@ Line 5: / Line 5: @@
 <math>S(X,Y) = \nabla_XY - \nabla'_XY</math>
-where <math>\nabla</math> si the [[Levi-Civita conncetion]] of <math>\R^n</math> and <math>\nabla'
+where <math>\nabla</math> si the [[Levi-Civita connection]] of <math>\R^n</math> and <math>\nabla'
 </math> is the Levi-Civita connection on <math>M</math>.