# Tangent space vs. Vector space

I'm not sure I fully understand the difference between these two terms when used in differential geometry/general relativity.

If I were to describe covariant differentiation to someone, I would say something like this:

"On a curved manifold (imagine a basketball), you could assume a tangent space at the point "P." At that point, the tangent space would be like placing a flat piece of paper on the ball at that point P.
If you move from point P to a new point "Q," you're no longer in that same tangent space and therefore need a "connection" to allow the differentiation to work for non-flat manifolds...yada yada..."

The reason I ask is because I did have a conversation with a fellow student. I was explaining what a Christoffel symbol was and why it was needed.
I remember saying "If you take the derivative, you're no longer in the same vector space" in the spot where I used tangent space above.

I also tried to explain the act of "getting a number" in a way that related to QM.
I said that in QM, you "get a number" by operating on something in the bra-ket notation.

"If you want a number for the energy, you can perform the operation
$$\langle \psi | H | \psi \rangle$$

In GR, it's not much different. If you want to "get a number," you combine vector and a dual vector. The contraction of the vector $$\upsilon^{\alpha}$$ with the dual vector $$f_{, \alpha}$$ in the form
$$\frac{df}{d\lambda}=f_{, \alpha}\upsilon^{\alpha}$$
gives you a scalar, i.e. "number."

I noticed myself using "tangent space" and "vector space" interchangeably in that conversation as well.

So......how exactly are they different?
Perhaps my "visual image" of the tangent plane on a curved manifold is what has me using them as equal?

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The tangent space is an example of a vector space, it's just a particular kind that you get from considering tangents to surfaces. A vector space is more general in that you just need to satisfy a bunch of axioms to qualify as one.

Another example of a vector space is the cotangent space, which is where dual vectors (a.k.a covectors or 1-forms) like df live. There are all kinds, kets like $|\psi\rangle$ that you mentioned also live in a vector space, but it's not a tangent space.

Fredrik
Staff Emeritus
Gold Member
The tangent space at some point in the manifold is a vector space.

The space of kets in QM isn't a tangent space of a manifold. It's just a vector space. (A Hilbert space to be more precise). The space of bras is another vector space, which is the dual space of the space of kets. If V is a topological vector space over a field F, the dual space V* is defined as the set of continuous functions from V into F.

The dual space of the tangent space at a point in a manifold is called the cotangent space at that point.

Edit: D'oh too slow.

Some of this has been covered already, but here's a little bit more detail...

Tangent spaces and cotangent spaces are particular kinds of vector space.

A vector space consists of a commutative group (also called an Abelian group), a field (in the abstract algebra sense, http://mathworld.wolfram.com/Field.html, rather than the "scalar/vector/tensor field" sense) and some function called scalar multiplication. Elements of the underlying set of the group are called the vectors of this vector space, elements of the underlying set of the field are called scalars. Scalar multiplication has two arguments (inputs), a scalar and a vector of this particular vector space; its value (output) is a vector of this particular vector space. To qualify as a vector space, this function must fulfill the following axioms. I'll use the familiar notation juxtaposing an italic letter denoting a scalar with a bold letter denoting a vector to mean scalar multiply the scalar on the left by the vector on the right:

$$(1) \enspace s(t\textbf{v})=(st)\textbf{v}$$

$$(2) \enspace (s+t)\textbf{v}=s\textbf{v}+t\textbf{v}$$

$$(3) \enspace s(\textbf{v}+\textbf{w})=s\textbf{v}+s\textbf{w}$$

$$(4) \enspace 1\textbf{v}=\textbf{v}$$

for all scalars s and t, and all vectors v and w.

Any structure you define which obeys these axioms, you can call a vector space. Its vectors might be finite or infinite sequences of numbers, they might be functions (as when tangent vectors are defined as directional derivative operators), they might be matrices, anything as long as your definitions of vector addition and scalar multiplication satisfy the above conditions. But...

In certain contexts where several different sorts of vector space are involved, one of them may be, by convention, thought of as primary, and its vectors refered to as "vectors" without qualification. So in this context, you may find tangent vectors called simply vectors, and contrasted with, say, cotangent vectors or tensors of a higher order, all of which are vectors in their own right with respect to their own vector spaces, whatever names they go by in practice. Hence your natural impression that the terms tangent space and vector space might be interchangeable.

As far as I can see, your expression "you're no longer in the same vector space" makes sense in the context, and is no different from saying you're not in the same tangent space, the important point there being not so much how precisely you name it but that there's a different one associated with each different point of the manifold.

Thanks for the replies. I know that the tangent space at the point P (from my first post) is a vector space, I wasn't sure when/how they differed.

So, if I'm following this correctly......then the following statements are correct (at least in their usage of "vector space" and "tangent space?"

"To perform vector space operations on the tangent space $$T_{p}(\mathcal{M})$$ , you first have to define a mapping for $$T_{p}(\mathcal{M}) \to (land \ of \ known \ vector \ operations)$$ so that the vector space operations can then be used on $$T_{p}(\mathcal{M})$$"

Or,
"The covariant one form $$f$$ on $$T_{p}(\mathcal{M})$$ is a function of the vectors/vector space at the point $$p$$. So if x is a vector in the vector space at p, then $$f$$ maps x into a "number."

....so, vector space is more general (which I guess I knew already), and the tangent space (which is just the space of surface tangents....much less general) is just a particular example of a vector space.....the vector space that considers the collection of vectors that are tangent to the surface at that point?

I apologize if I'm being too redundant (and it even looks that way to me) by pretty much just repeating what Fredrik and Tomsk already posted.
My school has ZERO relativity classes or research groups (or even profs with any GR knowledge), so I'm stuck with self studying this stuff in between major semesters at school.

I really appreciate the help, I think it cleared things up for me.

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Some of this has been covered already, but here's a little bit more detail...

Tangent spaces and cotangent spaces are particular kinds of vector space.

A vector space consists of a commutative group (also called an Abelian group), a field (in the abstract algebra sense, http://mathworld.wolfram.com/Field.html, rather than the "scalar/vector/tensor field" sense) and some function called scalar multiplication. Elements of the underlying set of the group are called the vectors of this vector space, elements of the underlying set of the field are called scalars. Scalar multiplication has two arguments (inputs), a scalar and a vector of this particular vector space; its value (output) is a vector of this particular vector space. To qualify as a vector space, this function must fulfill the following axioms. I'll use the familiar notation juxtaposing an italic letter denoting a scalar with a bold letter denoting a vector to mean scalar multiply the scalar on the left by the vector on the right:

$$(1) \enspace s(t\textbf{v})=(st)\textbf{v}$$

$$(2) \enspace (s+t)\textbf{v}=s\textbf{v}+t\textbf{v}$$

$$(3) \enspace s(\textbf{v}+\textbf{w})=s\textbf{v}+s\textbf{w}$$

$$(4) \enspace 1\textbf{v}=\textbf{v}$$

for all scalars s and t, and all vectors v and w.

Any structure you define which obeys these axioms, you can call a vector space. Its vectors might be finite or infinite sequences of numbers, they might be functions (as when tangent vectors are defined as directional derivative operators), they might be matrices, anything as long as your definitions of vector addition and scalar multiplication satisfy the above conditions. But...

In certain contexts where several different sorts of vector space are involved, one of them may be, by convention, thought of as primary, and its vectors refered to as "vectors" without qualification. So in this context, you may find tangent vectors called simply vectors, and contrasted with, say, cotangent vectors or tensors of a higher order, all of which are vectors in their own right with respect to their own vector spaces, whatever names they go by in practice. Hence your natural impression that the terms tangent space and vector space might be interchangeable.

As far as I can see, your expression "you're no longer in the same vector space" makes sense in the context, and is no different from saying you're not in the same tangent space, the important point there being not so much how precisely you name it but that there's a different one associated with each different point of the manifold.

Thank you very much for the detailed post, it's very helpful.
I think these posts have cleared up my confusion

Fredrik
Staff Emeritus
Gold Member
"To perform vector space operations on the tangent space $$T_{p}(\mathcal{M})$$ , you first have to define a mapping for $$T_{p}(\mathcal{M}) \to (land \ of \ known \ vector \ operations)$$ so that the vector space operations can then be used on $$T_{p}(\mathcal{M})$$"
This one looks weird to me. What's a "vector space operation"? Are you talking about the two functions involved in the definition of a vector space, i.e. addition and scalar multiplication? In that case, those functions have to be defined (once we have define which set to call the tangent space), but addition is a function from $T_pM\times T_pM$ into $T_pM$ and scalar multiplication is a function from $\mathbb R\times T_pM$ into $T_pM$. Functions from $T_pM$ into $\mathbb R$ don't enter the picture until it's time to define the cotangent space $T_pM^*$, which is a different vector space.

"The covariant one form $$f$$ on $$T_{p}(\mathcal{M})$$ is a function of the vectors/vector space at the point $$p$$. So if x is a vector in the vector space at p, then $$f$$ maps x into a "number."
This is better. I'd drop the word "covariant" though. A one-form is a member of $T_pM^*$, so it's a linear function from $T_pM$ into $\mathbb R$.

....so, vector space is more general (which I guess I knew already), and the tangent space (which is just the space of surface tangents....much less general) is just a particular example of a vector space.....the vector space that considers the collection of vectors that are tangent to the surface at that point?
This one is better. The tangent space at a point is the vector space of tangent vectors at that point. There are several different ways to define the tangent space at a point. It can be defined as a set of derivative operators at that point or as a set of equivalence classes of curves through that point. See this post and the one I linked to in it.

My school has ZERO relativity classes or research groups (or even profs with any GR knowledge), so I'm stuck with self studying this stuff in between major semesters at school.
It's not a bad idea to pick up a book on differential geometry. I really like Lee's "Introduction to smooth manifolds", but I think the concept of "tangent space" is covered even better in Isham's "Modern differential geometry for physicists".

It's not a bad idea to pick up a book on differential geometry. I really like Lee's "Introduction to smooth manifolds", but I think the concept of "tangent space" is covered even better in Isham's "Modern differential geometry for physicists".

That's a good idea. I seem to like learning the differential geometry more than the actual physics of GR so far.
I know I've heard Isham's book mentioned before too....I'll look for it at the library today.

Thanks again for the help!