I'm reading "tensor analysis on manifolds" by Bishop and Goldberg. I have taking a course in differential geometry i R^3. The course was held on Do Carmos book. Do carmo deffined the tangent at a point on a surface as all tangents to all curves on the surface going through that point (or something like that). This diffenition is pretty clear and easy to imagine.(adsbygoogle = window.adsbygoogle || []).push({});

In the other book it is defined as all darivations on all ****tions from the manifold at the point to R. Then they define the tangent of a curve in the manifold to be and operator. And then they define the tangent to the ith coordinate curve of gamma, and proof a theorem about that these coordinate tangents at a point m on the manifold is a basis for all tangents.

The proof are not that hard, but my problem is that i can't see that the two definitions are the same when i choose one of the "manifolds" we worked on in do carmo, fx. the unit sphere in R^3. I now the one is a operator and the other are vectors, but there most be a clear connection?

I've seen an example where they use R^2 as there manifolds, and then state that the directional derivative corospones to the operators. And then it is clear that given and directional derivative in the direction v lets call it Dv, then i have the operator and the if i define a function f(Dv)=v then i get all tangent vectors to all points in R^2.

But somehow i can't find the connection when my surface gets more complicated, like the unit sphere. I can see that the operator is connected to the tangent of curves through m, but it is not so clear to me, in my mind there most be a function (possible a bijection) on the "tangent operators" such that i get all tanget vectors, when my manifold is so simple like the unit sphere that a "normal" tangent vector/plane makes sence. My point is that if it makes sence to define the tangents like operaors, I most have just as much information as if I had defined it the "usual" way (of cause the new defintion makes more sence when i have obscure manifolds where tangentplanes are not clear, but in R^3 there most be a clear connection in my head).

I hope someone can understand what my problem is, anbd have the commitment to read the whole post (it got a bit long, sorry). And by the way, i'm sorry for the bad englsih.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Tangent space, derivation defifinition

Loading...

Similar Threads for Tangent space derivation | Date |
---|---|

I Definition of tangent space: why germs? | Oct 2, 2017 |

A Tangent bundle: why? | Mar 3, 2017 |

I Directional Derivatives and Derivations - Tangent Spaces | Feb 23, 2016 |

Tangent Spaces of Parametrized Sets - McInerney, Defn 3.3.5 | Feb 19, 2016 |

Tangent spaces and derivations | Jun 29, 2007 |

**Physics Forums - The Fusion of Science and Community**