First fundamental form and squared arc length element

In summary: Is there an equivalent definition for the fundamental forms for higher dimensions or not?In summary, the first and second fundamental forms are defined for Riemannian manifolds, with the first fundamental form being a positive definite inner product on the tangent space and the second fundamental form being the pullback of the metric by the immersion. While the terminology of the first fundamental form is outdated, it is still used in textbooks about the differential geometry of surfaces in R³. An immersion is a smooth mapping of one manifold into another such that the derivative is injective at every point.
  • #1
Vasileios
6
0
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 [tex]ds^2[/tex]


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

Thanks
Vasileios
 
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  • #2
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.
 
  • #3
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
[tex]I(v,w)= v^T[E F ; F G]w[/tex]
where the coefficients can be written by the Riemannian metric
[tex](g_{ij})= [E F ; F G][/tex]

From that the squared arc length element follows:

[tex]ds^2=Ed_v^2+2Fd_vd_w+Gd_w^2[/tex]

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
[tex]ds^2=\sum g_{ij}d_id_j[/tex] ?

Is it as simple as having 6 coefficients instead of 3?
 
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  • #4
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.
 
  • #5
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 [itex]\nabla[/itex] is the Levi-Civita connexion on (M,g) and N is an immersed submanifold, then for vector fields X, Y on N,
[itex]II(X,Y)=(\nabla_XY)^{\perp}[/itex], where X,Y are arbitrary smooth extensions of X and Y to M, and [itex]\perp[/itex] means projection onto the normal bundle of N.
 
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  • #6
quasar987 said:
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?
 
  • #7
7thSon said:
What part of this terminology is outdated?

First fundamental form.

7thSon said:
And what are the more modern terms?

Something like

lavinia said:
Riemannian metric that it inherits from the ambient manifold.


7thSon said:
Also, what is the proper, two sentence definition of the immersion?
What?
 

1. What is the first fundamental form?

The first fundamental form is a mathematical tool used to measure the intrinsic properties of a curved surface. It consists of three coefficients that describe the local geometry of the surface at a given point.

2. How is the first fundamental form used to calculate the squared arc length element?

The first fundamental form can be used to calculate the squared arc length element by taking the dot product of two tangent vectors on the surface and then taking the square root of the result. This gives the length of the infinitesimal arc on the surface.

3. What is the significance of the squared arc length element?

The squared arc length element is important because it allows us to measure distances on a curved surface in a way that is independent of the parameterization or coordinates used. This is essential for accurately describing and studying the geometry of curved surfaces.

4. Can the first fundamental form and squared arc length element be used for any type of surface?

Yes, the first fundamental form and squared arc length element can be used for any type of surface, whether it is smooth, continuous, or even has singularities or discontinuities. As long as the surface is defined by a set of coordinates, the first fundamental form can be calculated and used to measure distances on the surface.

5. How is the first fundamental form related to the curvature of a surface?

The first fundamental form is closely related to the curvature of a surface, as it provides information about the local geometry of the surface at a given point. The coefficients in the first fundamental form can be used to calculate the Gaussian and mean curvature, which are measures of the overall curvature at that point.

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