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Heisenberg1993
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Why is i that two cones connected at their vertices is not a manifold? I know that it has to do with the intersection point, but I don't know why. At that point, the manifold should look like R or R2?
But it doesn't, so it is not a manifold.Heisenberg1993 said:At that point, the manifold should look like R or R2?
It doesn't look like either. There is no smooth map from an open set of R nor R2 that maps to that region.Heisenberg1993 said:That's exactly my question, why it doesn't look like which (R or R2)?
Heisenberg1993 said:That's exactly my question, why it doesn't look like which (R or R2)?
fresh_42 said:...Now there is no continuous way to make one disc or ball out of it, because you lose uniqueness in this point.
Yes, I indeed emphasized the wrong property as the bijection is the crucial point.lavinia said:You can continuously ( in fact smoothly) map an open double cone onto an open interval in ##R##. You can then continuously map the interval onto an open subset of the plane.
Orodruin said:It doesn't look like either. There is no smooth map from an open set of R nor R2 that maps to that region.
Fine, as a physicist, I mainly just consider smooth manifolds.lavinia said:There is no smooth map from an open set in the plane (or the real line) onto a region of the vertex of a single cone either. Yet a single cone is a manifold.
Orodruin said:Fine, as a physicist, I mainly just consider smooth manifolds.
micromass said:The single cone can be very easily be considered a smooth manifold. A double cone can never be a smooth manifold. That's the difference.
This is because a manifold is a space that locally resembles Euclidean space, meaning it should be smooth and have a well-defined tangent space at every point. However, at the point where the two cones are connected, the surface becomes non-smooth and the tangent space is not well-defined, making it not a manifold.
Yes, two cones connected at their vertices can still be considered a surface. However, it is not a manifold since it does not locally resemble Euclidean space.
A manifold is a smooth surface that locally resembles Euclidean space, while a non-manifold surface is not smooth and does not have a well-defined tangent space at every point. A manifold can be described using mathematical equations, while a non-manifold surface cannot.
Yes, a real-life example of this is the shape of a saddle. When viewed from above, it may look like two cones connected at their vertices, but the surface is not smooth and does not locally resemble Euclidean space.
The non-manifold property of two cones connected at their vertices affects its properties by making it non-smooth and not well-defined at the point of connection. This can impact its curvature, area, and other geometric properties, making it difficult to analyze and describe mathematically.