Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I've just read and I think I have understood the following result :

If we were to geodesically transport all points of a small 2D surface, so small that it would be flat for all purposes, in a direction vertically above it, and if this surface belongs in an arbitrary 3D manifold, then in general both its area and its shape will change. It is impossible to achieve a change in shape without its area to dwindle or expand simultaneously. And this is a property of the so-called Weyl tensor.

I can somehow visualize how this is happening without recourse to advanced maths, but what a great result this is!

In a 4D manifold, it is also true that the object can retain its volume even if the tidal forces (curvature) change its shape, but not in any lower dimension than 4.

I have nothing more to ask, just to verify if this is true. What a great result! If it is true, it is a moment of revelation and enlightening for me.

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

Dismiss Notice

Join Physics Forums Today!

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

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

# Geodesic transport of a small 2D surface on a 3D manifold.

Loading...

Similar Threads - Geodesic transport small | Date |
---|---|

A Commutator of covariant derivative and D/ds on vector fields | Mar 15, 2018 |

A Showing E-L geodesic def and covariant geodesic def are same | Mar 6, 2018 |

A Constant along a geodesic vs covariantly constant | Feb 27, 2018 |

Fermi-Walker transport geodesic | Jul 11, 2011 |

Parallel transport and geodesics | Oct 16, 2009 |

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