0

Differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics

Yes, but I don't understand which method is used, I need to see an example.

How is a tensor defined on the tangent space in the point associated with each point of a differentiable manifold, for example an open of the Euclidean space $R^n%$?

Kann
  • 3

1 Answers1

0

We can define a vector field on a differentiable manifold, where the vector at each point $\mathbf{P}$ is an element of the tangent space $\mathbf{T_P}$ at $\mathbf{P}$.

As well as a tangent space $\mathbf{T_P}$, there is also a cotangent space $\mathbf{T^*_P}$ at each point $\mathbf{P}$. This is the space of linear maps from $\mathbf{T_P}$ to $\mathbb{R}$ (or from $\mathbf{T_P}$ to $\mathbb{C}$ if we are considering a complex manifold).

Once we have a tangent and cotangent space at each point then we can define a tensor field where the tensor at each point is a linear map from some product of copies of $\mathbf{T_P}$ and $\mathbf{T^*_P}$ to $\mathbb{R}$ (or $\mathbb{C}$ ). A tensor field may represent some intrinsic attributes of the manifold (the Riemann curvature tensor field, for example) or it may represent extrinsic physical attributes (the stress-energy tensor field, for example).

gandalf61
  • 15,326
  • when you say "we can define" what do you mean practically? How do you define, for example, a vector space, that is, how do you choose to represent it "on"? With matrices? How will you represent your differentiable manifold on which you go to "define" a vector field? – Kann Jul 26 '19 at 15:26
  • @Kann A vector field is a collection of objects called vectors, one for each point in a manifold, where the vector associated with each point is an element of the tangent space at that point. That is the definition of a vector field. We can multiply a vector field by a scalar or add two vector fields by performing that operation in the tangent space at each point in the manifold. This creates a new vector field. Does that make sense ? – gandalf61 Jul 26 '19 at 15:41