Prove $$T_{p}M$$ is a vector space with the axioms

In summary, we have defined a tangent space at a point $$p \in M$$ as the set of all tangent vectors at that point, denoted by $$T_pM$$. To prove that it is a vector space, we must show that it satisfies the axioms of a vector space. These axioms include closure under addition and scalar multiplication, commutative and associative properties, an additive identity and inverse, and a scalar identity. We can define the tangent vectors as operators on differentiable functions at $$p$$ and use the coordinate directions and partial derivatives to show the axioms hold.
  • #1
Delong66
4
0
Suppose M is a manifold and $$T_{p}M$$ is the tangent space at a point $$p \in M$$. How do i prove that it is indeed a vector space using the axioms:
Suppose that u,v, w $$\in V$$. where u,v, w are vectors and $$\V$$ is a vector space

$$u + v \in V \tag{Closure under addition}$$

$$u + v = v + u \tag{Commutative property}$$

$$u + (v+w)=(u+v)+w \tag{Associative property}$$

V has a zero vector 0 such that for every $$\u \in \V$$, $$u+0=u$$. $$\tag{Additive identity}$$

For every $$u \in V$$, there is a vector in V denoted by −u such that u+(−u)=0. $$\tag{Additive inverse}$$

Now let's also assume that $$c,d \in \mathbb R$$

$$cu \in V \tag{Closure under scalar multiplication}$$

$$c(u+v)=cu+cv \tag{Distributive property}$$

$$(c+d)u=cu+du \tag{Distributive property}$$

$$c(du)=(cd)u \tag{Associative property}$$

$$1(u)=u \tag{Scalar identity}$$
 
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  • #2
You listed the axioms of a vector space, but how do you define the tangent space? What does an element of ##T_pM## look like?
 
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  • #3
Definition 1: Suppose M a differentiable manifold and $$p\in M$$.
A funtion $$f:M \rightarrow \mathbb{R}$$ is differentiable at $$p \in M$$ iff $$\exists U_p \subset M$$ : $$f:U_p \rightarrow \mathbb{R}$$ is differentiable.

Definition 2:Dp ={set of all differentiable functions at p in M}

Definition 3: Suppose M a differentiable manifold and $$p\in M$$.
A tangent vector at $$p\in M$$ is an operator $$v:D_p \rightarrow \mathbb{R} \quad : \quad f \mapsto v(f)$$ with
i) $$v(mf+ng)=mv(f)+nv(g) \quad \forall f,g \in D_p$$
ii)$$v(fg)=v(f)g(p)+f(p)v(g) \quad \forall f,g \in D_p$$
\
Definition 4: TpM={set of all tangent vectors at p in M}
 
  • #4
Definitions 1 and 2 define the differentiability at points of ##M,## i.e. differential functions, not their tangents.

Definition 3 defines one tangent vector, definition 4 defines the set of all tangent vectors. However, it does not tell anything about the relation between two such operators ##v## and ##w.## We now have a set, nothing more. How do we make this set a vector space?

We need to define ##v+w## so we set: ##(v+w)(f):=v(f)+w(f)## and ##(\alpha v)(f):=\alpha \cdot v(f).##
This makes the set of derivations, the operators, a linear space. The axioms follow automatically.

If you want to calculate something, then consider curves on ##M## through ##p## and calculate actual tangent vectors and show that ##D_p(v+w)(f)=D_p(v)(f)+D_p(w)(f).## You can also use the coordinate directions and partial derivatives:
$$
D_p(v)(f)=\sum_{i=1}^n\, v_i\, \frac{\partial f}{\partial x_i}(p)
$$
This could also help (##p=x_0, J=D, J## for Jacobi matrix)
https://www.physicsforums.com/insights/pantheon-derivatives-part-ii/
 
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1. What is the definition of a vector space?

A vector space is a mathematical structure that consists of a set of objects, called vectors, and a set of operations, called scalar multiplication and vector addition. These operations must satisfy a set of axioms, or rules, in order for the set to be considered a vector space.

2. What are the axioms of a vector space?

The axioms of a vector space include closure under scalar multiplication and vector addition, associativity and commutativity of vector addition, distributivity of scalar multiplication over vector addition, existence of an additive identity element, and existence of additive inverses for each vector. These axioms ensure that the set of vectors behaves consistently and can be manipulated algebraically.

3. How do you prove that $$T_{p}M$$ is a vector space?

To prove that $$T_{p}M$$ is a vector space, we must show that it satisfies all of the axioms of a vector space. This can be done by demonstrating that the set of tangent vectors at a point p in a manifold M satisfies the closure properties, associativity and commutativity, distributivity, and existence of identity and inverse elements. Additionally, we must show that the set is closed under scalar multiplication and vector addition.

4. What is the significance of proving that $$T_{p}M$$ is a vector space?

Proving that $$T_{p}M$$ is a vector space is important because it allows us to use the powerful tools and techniques of linear algebra to study the behavior of tangent vectors on a manifold. This can help us better understand the geometry and topology of the manifold, as well as make calculations and predictions about the behavior of objects moving along the manifold.

5. Are there any exceptions to the axioms of a vector space?

Yes, there are exceptions to the axioms of a vector space. For example, in non-Euclidean geometries, the axioms of vector spaces may not hold true. Additionally, in quantum mechanics, the concept of a vector space is extended to include complex numbers and the axioms are modified accordingly. However, for the purposes of studying tangent vectors on a manifold, the traditional axioms of a vector space still hold true.

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