# I Describing 3d Manifold Objects as a Hypersurface

1. Dec 21, 2017

### decentMO

Hello, an engineer here that is trying to understand more in topology for some mechanics ideas. It seems to me that any closed (regular) manifold in 3d Euclidean space should be able to be describe uniquely by the combination of the following 1) volume [1 real], 2) centroid [3 reals], and 3) second moments of area [9 reals cast to 7 since third axis is orthogonal to cross of 1st and 2nd]. The surface is defined of the object as C0 or higher. Can anyone give any direction as to confirming this conjecture? Is there an 11-dimensional hypersurface that is known that defines the permissible (manifold) objects in R3?

2. Dec 25, 2017

### WWGD

In the standard setup you can only embed $n-$ manifolds in $n-$ dimensional space or lower. By the same setup , a hypersuface has codimension 1 , i.e., lives in dimension of the surface+1.

3. Dec 25, 2017

### mathwonk

i think he is asking whether the space parametrizing all solids? in 3 space can itself be realized as an 11 dimensional manifold in I guess 12 space. This is of course not a topological question since he is using volume and area, but a question in differential geometry.

4. Dec 25, 2017

### decentMO

Yes, that's it. Thank you for clarifying for me and my apologies as I am not familiar with the differences between topology and differential geometry. Would it be helpful to move this question to a different forum? I'm want to do some reading on this subject but I'm not sure where to start in the right direction, so if you have salient reading recommendations, I'm open to suggestions. Thanks!

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