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kent davidge
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How do you know if two given points on a manifold have the same tangent space? Checking if a vector does not change when transported from one point to the other is enough?
What do you mean by the same? They cannot be the same at different points. They are isomorphic, as e.g. on a n-manifold they both are isomorphic to ##\mathbb{R}^n##. So again, what is supposed to mean "equal"?kent davidge said:How do you know if two given points on a manifold have the same tangent space? Checking if a vector does not change when transported from one point to the other is enough?
oh that makes sensefresh_42 said:They cannot be the same at different points. They are isomorphic, as e.g. on a n-manifold they both are isomorphic to ##\mathbb{R}^n##
I mean, if we consider the m-dimensional manifold to be ##\mathbb{R}^m## itself, we can find a (global) coordinate system where the basis don't change, namely a Cartesian coordinate system. In that coordinate system, a vector (not a vector field, so constant components) will not change at all if we move from one point to another. That's what we usually do geometrically when we e.g. drag the arrows around in the plane, right?fresh_42 said:What do you mean by the same? So again, what is supposed to mean "equal"?
Just to add that this motivates the concept of connections, which, well, connect tangent spaces at different points. In euclidean n-space, the isomorphism is natural, but not so in general manifolds.kent davidge said:oh that makes sense
I mean, if we consider the m-dimensional manifold to be ##\mathbb{R}^m## itself, we can find a (global) coordinate system where the basis don't change, namely a Cartesian coordinate system. In that coordinate system, a vector (not a vector field, so constant components) will not change at all if we move from one point to another. That's what we usually do geometrically when we e.g. drag the arrows around in the plane, right?
Now for me it seems unecessary to continue saying that the points in ##\mathbb{R}^m## have different tangent spaces, as we can do what I just described above. From this follows my question in post #1, if checking the constancy of a vector is a sufficient condition.
A tangent space at a point is a mathematical concept that describes all possible directions or velocities that a curve or surface can have at that point. It is a vector space that is tangent to the curve or surface at that point.
A tangent space at a point is specific to that point and describes all possible directions or velocities at that point. A tangent space at a different point would describe all possible directions or velocities at that different point, which may be different from the first point.
Tangent spaces at different points are important because they allow us to analyze the behavior of a curve or surface at different points. They also help us understand the local geometry of a curve or surface, which is crucial in many areas of mathematics and physics.
Tangent spaces at different points are related through the concept of differentiability. If a curve or surface is differentiable at a point, then the tangent spaces at nearby points will be similar. As the points get closer together, the tangent spaces will become more and more similar.
Yes, the dimension of a tangent space at a point can be different from the dimension of a tangent space at a different point. The dimension of a tangent space depends on the dimension of the original space and the number of independent variables needed to describe the curve or surface at that point. Two points on a curve or surface may require different numbers of variables, resulting in different dimensions for the tangent spaces at those points.