Which theorem? Determining distortion of a 3d-object in 4D

[Spacie]

I am looking for a theorem that states approximately the following:

An n-dimensional object, while appearing perfectly regular within the n-dimensional space to which it belongs, can actually be bent or distorted in (n+1) dimensions.​

Please forgive my ignorance of the proper terms. I'm a newbie and want to learn.

Here is the my take on the problem on a simple model of Flatland suspended in 3 dimensions, exemplified with a sheet of paper in our familiar world (everything is examined in the framework of a hypothetical 3D Euclidean space) : In the simplified case of Flatland, the 3rd dimension emerges orthogonally from its 2D sheet. If I sample 2 points on Flatland's 2D plane, by drawing a line through each point into the 3rd dimension, orthogonal to the plane, and then compare the angle between the resulting 2 lines, then, if this angle is 0, it means that they are parallel to each other and the area between these 2 sampling points is flat. If not, this means that the 2D plane is curved in 3D between these 2 points.

That part was simple.

Now, in 4D, a 3D object has 3 bounding planes, orthogonal to each other (XY, XZ and YZ). Looking at this 3D object from within the same 3D (i.e. playing with a cube in our familiar 3D world), with nothing distorted:

XY ⊥ XZ, XY ⊥YZ , XZ ⊥YZ
and if I draw a line a ⊥ XY, line b ⊥ XZ and line c ⊥ YZ,
these 3 lines, a b and c, are not parallel to each other. (they are in fact ⊥ to each other).Now looking at 3D object from the 4th dimension, no distortions:

again I draw 3 lines orthogonal to XY, XZ and YZ, but this time I draw them into the 4th dimension:

a ⊥ XY, b ⊥ XY, c ⊥ YZ
in the 4th dimension, these 3 lines, a b and c, are parallel to each other.


In other words, from the point of view of the 4th dimension, the 3 planes XY, XZ and YZ belong to the same 4-plane -?

.. and if it is not flat (determined by similar test to Flatland above) then the 3-dimensional object in question is distorted in 4 dimensions -?Intuitively this appears evident to me, but.. instead of me re-inventing this bicycle, I'd like to avail myself of an already existing, properly formulated topological theorem. It got to be out there. What's it called and where can I find it?

Thank you for your feedback :)

 

[pat_connell]

The theorem you are looking for is known as the Gauss-Bonnet Theorem. It states, in part, that the curvature of an n-dimensional object is related to its topology and the Euler characteristic of the object. This theorem is described in greater detail in many textbooks on differential geometry and topology.