What is the 2nd Fundamental Form and how does it relate to geometry in 3-space?

[Reality_Patrol]

Can anyone provide an intuitive explanation for what the 2nd fundamental form is (maybe in 3-space geometry)? How does it relate to the 1st fundamental form??

Perhaps surprisingly I've read over (& understood fairly well) an advanced explanation, but it doesn't provide the kind of intuitive understanding I'm seeking. Thanks in advance!

 

[mathwonk]

there are only two concepts in differential geometry, length and curvature, so probably the first fundamental form is length and the second is curvature?

 

[Reality_Patrol]

Uhh, not exactly...the "advanced" explanation of the 2nd fundamental form I found described it as an "intrinsic" metric for determining the infinitesimal distance to a parallel surface (embedded in a n+1) space. So, I guess you could say that it's a "extrinsic length" but I don't get what it is within the n-space itself (i.e. intrinsically). I'm sure it has an geometrical interpretation within it's own n-space. I would like to find that alternative interpretation and understand it in familiar 3-space.

 

[robphy]

Maybe you are referring to the "extrinsic curvature", which can be interpreted in terms of the lie derivative of the hypersurface metric along the unit normal.

 

[mathwonk]

well i just googled second fundamental form, and i was right, the first fundamental form is the dot product of two vectors, hence equivalent to the length or metric,

and the second fundamental form is an operator on two vectors, dotting the second one with, as robphy says, the extrinsic curvature of the first vector, i.e. the derivative of the normal vector to the surface in the direction of the given vector.

I still like my "zen" approach to answering your question. what else could it be?: there are no other concepts in the subject. actually that is not right, there are as many concepts as there are derivatives. so is suppose the "third" fundamental form is the torsion.

i'll go look it up and see.

i seem to have given the namers too much credit, the third form is just a product of two second forms.

 

[Reality_Patrol]

Mathwonk,

Thanks for the reply, it certainly was a help. Still see some comments below:

well i just googled second fundamental form, and i was right, the first fundamental form is the dot product of two vectors, hence equivalent to the length or metric,

Equivalent dimensionally (in terms of units) but not geormetrically.

and the second fundamental form is an operator on two vectors, dotting the second one with, as robphy says, the extrinsic curvature of the first vector, i.e. the derivative of the normal vector to the surface in the direction of the given vector.

Ahh, but what are these initial two vectors that so uniquely define the second fundamental form? Your definition suggests any 2 vectors, that's not the case. Also, I'm trying to understand it in terms of intrinsic geometry - your description involves extrinsic concepts. Maybe there is no purely intrinsic view.

I still like my "zen" approach to answering your question. what else could it be?: there are no other concepts in the subject...

OK, I'll go along with your zen approach. So it's a length? What does that really mean geometrically since it's certainly not the "distance between two points" (as the first fundamental form gives that)??