First fundamental form and squared arc length element

[Vasileios]

First of all hello,
I am new to this forum and I decided to join in order to exchange some information with other members that are more knowledgeable than me in the area of diff. geometry.

My background is computer science but I am not a student. I am only now starting to learn about diff. geometry (and in particular information geometry which is my interest). So I my questions are going to be mostly basic. Also maybe sometimes my use of terminology is not 100% and i apologise for that, but it will become better in time :)


So the first question I would like to ask is the connection between the first fundamental form and the sq. arc length element $$ds^2$$


It seems to me that the first (and second) fundamental forms are only defined for 2d manifolds in $$R^3$$ whereas the $$ds^2$$ as the sum of $$g_{ij}$$ is arbitrarily dimensional. So my question is, is there an equivalent definition for the fundamental forms for higher dimensions or not?

Thanks
Vasileios

 

[lavinia]

For any Riemannian manifold there is by definition a first fundamental form - a positive definite inner product on the tangent space. For a manifold embedded as a hypersurface in another Riemannian manifold you have a second fundamental form.

 

[Vasileios]

Hi thanks for the reply.

Ok I am a bit confused here.
The definition for the first fundamental form I have read about somehow is only defined for manifolds in R3

so
$$I(v,w)= v^T[E F ; F G]w$$
where the coefficients can be written by the Riemannian metric
$$(g_{ij})= [E F ; F G]$$

From that the squared arc length element follows:

$$ds^2=Ed_v^2+2Fd_vd_w+Gd_w^2$$

Suppose then I have a 3-manifold in R^4.
Can I still express the arc length element as a function of the first fundamental form, instead of
$$ds^2=\sum g_{ij}d_id_j$$ ?

Is it as simple as having 6 coefficients instead of 3?

 

[lavinia]

Well maybe you are right that the first fundamental is only defined for manifolds embedded in others. It is just the Riemannian metric that it inherits from the ambient manifold. For a parametrized 3 manifold it would be expressed in terms of pairwise products of the 3 parameter variables. I believe the second fundamental form can be defined for any hypersurface of a Riemannian manifold. In fact it probably does not require a submanifold of codimension 1. I will check this.

 

[quasar987]

The first fundamental form (written "I(X,Y)") can be defined for any immersed submanifold of a riemannian manifold. It is just the pullback of the metric by the immersion. However, this terminology is outdated. The only place it seems to stick is in textbooks about the differential geometry of surfaces in R³ as a shockwave of the influence of Gauss (?).

However, the second fundamental form, which also can be defined on any immersed submanifold of a riemannian manifold, is still called the second fundamental form and written "II(X,Y)".

If $\nabla$ is the Levi-Civita connexion on (M,g) and N is an immersed submanifold, then for vector fields X, Y on N,
$II(X,Y)=(\nabla_XY)^{\perp}$, where X,Y are arbitrary smooth extensions of X and Y to M, and $\perp$ means projection onto the normal bundle of N.

 

[7thSon]

The first fundamental form (written "I(X,Y)") can be defined for any immersed submanifold of a riemannian manifold. It is just the pullback of the metric by the immersion. However, this terminology is outdated. The only place it seems to stick is in textbooks about the differential geometry of surfaces in R³ as a shockwave of the influence of Gauss (?).

What part of this terminology is outdated? And what are the more modern terms?

Also, what is the proper, two sentence definition of the immersion?

 

[quasar987]

What part of this terminology is outdated?

First fundamental form.

And what are the more modern terms?

Something like

Riemannian metric that it inherits from the ambient manifold.

Also, what is the proper, two sentence definition of the immersion?

What?