# Opposite "sides" of a surface - Differential Geometry.

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• thehangedman
In summary, the conversation discussed the concept of two-sidedness in differential geometry and how it differs between surfaces and higher dimensional manifolds. It was determined that the triviality of the normal bundle is necessary and sufficient for a surface to be two-sided. This concept is not related to affine connections or the difference between covariant and contravariant vectors.
thehangedman
How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while the other remained the same? Would the affine connection change at all? Is this at all related to the differences between a covariant and contra variant vector?

Ultimately, as I understand it, a surface S embedded in a manifold M is one-sided if S has a trivial normal bundle, which is equivalent to being co-orientable, meaning sort of being orientable within the ambient space. A coorientation is a smooth assignment of a (unit) normal vector field at each point of S .I believe this is equivalent to having a continuous non-tangent vector field . I don't understand what the question about the connection or the coordinates is about. Of course, normality is itself a function of the choice of Riemannian metric, but I don't get the question. As I understand it, a "side" is given by a choice of coorientation, tho maybe someone can chime in and add something here.

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WWGD said:
Ultimately, as I understand it, a surface S embedded in a manifold M is one-sided if S has a trivial normal bundle, which is equivalent to being co-orientable, meaning sort of being orientable within the ambient space. A coorientation is a smooth assignment of a (unit) normal vector field at each point of S .I believe this is equivalent to having a continuous non-tangent vector field . I don't understand what the question about the connection or the coordinates is about. Of course, normality is itself a function of the choice of Riemannian metric, but I don't get the question. As I understand it, a "side" is given by a choice of coorientation, tho maybe someone can chime in and add something here.

Trivial normal bundle is correct. For a smooth compact hypersurface,##S##, with trivial normal bundle, a tubular neighborhood is diffeomorphic to ##S##x##[-1,1]##. The two pieces, ##S##x##[-1,0]## and ##S##x##[0,1]## define the two sides.

On the other hand, if the normal bundle is not trivial, then the hypersurface is not two sided. So trivial normal bundle is both necessary and sufficient.

A hypersurface of Euclidean space is always two sided because it separates Euclidean space into two disjoint pieces. One of these pieces is a bounded domain and intuitively can be thought of as the interior region defined by the hypersurface. One might generalize this to compact manifolds ,##S##, that are boundaries of one higher dimensional manifolds,##M##. In this case, normal bundle to ##S## is automatically trivial and ##M## can be thought of as the interior region defined by ##S##. (This does not mean that the boundary manifold has to be orientable. For instance, the Klein bottle is the boundary of a 3 dimensional manifold. ). This construction does not work in general, because there are many manifolds that are not boundaries e.g. the projective plane.

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thehangedman said:
How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while the other remained the same? Would the affine connection change at all? Is this at all related to the differences between a covariant and contra variant vector?

A compact surface in 3 space divides 3 space into two disjoint regions, one bounded the other unbounded. These two regions are the two sides that you were thinking of. A compact hypersurface of a higher dimensional Euclidean space will also divide the Euclidean space into two disjoint regions.

AS WWGD has rightfully pointed out, two sidedness really has to do with whether there are well defined opposite directions normal to the hypersurface (trivial normal bundle) and this condition generalizes the case of hypersurfaces of Euclidean space. A hypersurface of a general manifold even if it is two sided may not separate the manifold into two disjoint pieces.

- Two sidedness has nothing to do with affine connections on the manifold. It is a question of Differential Topology, not of Differential Geometry. Why do you think this has anything to do with affine connections?
- The question about whether two sidedness has anything to do with the difference between covariant and contravariant vectors doesn't seem to make sense. Can you explain what you were thinking of?

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EDIT: an impolite exchange has been removed and the thread will remain closed.

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## What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves, surfaces, and other geometric objects in higher dimensional spaces. It uses techniques from calculus and linear algebra to study the intrinsic and extrinsic properties of these objects.

## What does "opposite sides" of a surface refer to in differential geometry?

In differential geometry, "opposite sides" of a surface refer to two sides of a surface that are parallel to each other and are related by a symmetry transformation. This concept is used to study the symmetries and transformations of surfaces in higher dimensional spaces.

## What are some examples of surfaces with opposite sides in differential geometry?

Some examples of surfaces with opposite sides in differential geometry include a cylinder, a cone, and a torus. These surfaces have two parallel sides that can be related by a symmetry transformation, such as translation or rotation.

## How does differential geometry relate to other fields of science?

Differential geometry has applications in many fields of science, including physics, engineering, and computer graphics. It is used to study the shape and curvature of objects in space, and has applications in fields such as general relativity, computer vision, and robotics.

## What are some real-world applications of understanding "opposite sides" of a surface in differential geometry?

Understanding "opposite sides" of a surface in differential geometry has many practical applications. For example, it is used in computer graphics to create realistic 3D models of objects with symmetric properties. It is also used in physics to study the symmetries and transformations of objects in space, and in engineering to design structures that can withstand different types of stress and forces.

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