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--- ebsd2021:tema1 [2021/08/20 20:09] – escola
+++ ebsd2021:tema1 [2021/08/31 22:34] (current) – escola
@@ Line 1: / Line 1: @@
-Tema1-Minicurso Knieper
+**Tema 1: The geodesic flow, the tangent bundle $TTM$, Jacobi fields**
@@ Line 59: / Line 59: @@
 for $\xi\in (TTM)_v$, $v\in TM$, we will define a decomposition of $\xi$ into its horizontal and vertical components.
-Since $TM$ is fibered by vertical subspace, it is easy to define vertical vectors in $TTM$
+Since $TM$ is fibered by vertical subspaces, it is easy to define vertical vectors in $TTM$
 (and horizontal components).
 Consider the natural projection $\pi:TM\to M$, $\pi(v)=\gamma_v(0)$, the basis point of $v$.
-Then $D\pi$ measures how much curves in $ TM$ move on $M$ (with respect to their base points).
+Then $d\pi$ measures how much curves in $ TM$ move on $M$ (with respect to their base points).
-//Vertical vectors:// $\xi\in TTM$ is called a //vertical vector// if $\xi\in {\rm ker}(D\pi)$.
+//Vertical vectors:// $\xi\in TTM$ is called a //vertical vector// if $\xi\in {\rm ker}(d\pi)$.
 Denote the vertical vectors at $(TTM)_v$ by $V(v)$.
-Since $D\pi$ is surjective, each $V(v)$ has dimension $n$ and so
+Since $d\pi$ is surjective, each $V(v)$ has dimension $n$ and so
 the space of vertical vectors is a subbundle of $TTM$ of dimension $n$.
-By definition, vertical vectors are those $\xi=X'(0)$ tangent to curves $X(t)$ whose basepoint does not move infinitesimally at $t=0$. As we will see, $D\pi(\xi)$ will represent the horizontal component of $\xi$,
+By definition, vertical vectors are those $\xi=X'(0)$ tangent to curves $X(t)$ whose basepoint does not move infinitesimally at $t=0$. As we will see, $d\pi(\xi)$ will represent the horizontal component of $\xi$,
 so that $\xi$ is vertical iff its horizontal component is zero.