Clarification on notation regarding fields, forms, and exterior algebra

Question

Sorry if I've missed something quite obvious, but I can't seem to find a clear source for notation.

1. $\Omega^p(T^*M)$ is the common notation I've seen for the space of $p$-forms on the cotangent bundle $T^*M$. Is $\Omega^p(T^*M)=\bigwedge^p(T^*M)$ a correct statement, or is there some other step I'm missing (where $\bigwedge(\cdot)$ is the exterior algebra)?
2. What does $\Gamma(\cdot)$ mean when applied to one of these spaces (e.g., $\Gamma\bigwedge^p(T^*M)$)? I've seen it mean 'all the fields on the space', but this doesn't seem consistent everywhere, and feels somewhat vague.
3. What space does an $(n,m)$-rank tensor belong to? Is it the direct sum $\bigwedge^n(TM)\oplus\bigwedge^m(T^*M)$?

Thank you for your time, and again, sorry if this is made obvious in some text, I've not seen it.

Answer 1

Welcome to differential geometry where the notation is always an issue :)

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At the Level of Vector Spaces.

First, let's clarify things on the level of vector spaces. So, let $V$ be a real vector space.

• If $p\geq 1$ is an integer then by $\bigwedge^p(V)$ we mean the $p^{th}$ exterior power of the vector space $V$. Note that this is a different space from $\bigwedge^p(V^*)$ (though sometimes people omit the $^*$ and some confusion may arise if consulting several sources). For $p=1$, $\bigwedge^1(V):=V$.

• Next, given integers $r,s\geq 0$, an $(r,s)$ tensor on $V$ is by definition an element of $T^r_s(V):= \underbrace{V\otimes \cdots \otimes V}_{\text{$r$ times}}\otimes \underbrace{V^*\otimes \cdots\otimes V^*}_{\text{$s$ times}}$. For finite-dimensional $V$, this is isomorphic to the space of $(r+s)$-multilinear maps $\underbrace{V^*\times\cdots \times V^*}_{\text{$r$ times}}\times \underbrace{V\times\cdots \times V}_{\text{$s$ times}}\to \Bbb{R}$ (yes the $r,s$ and location of $^*$ has switched, this is no typo).

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At the Level of Vector Bundles

Let $(E,\pi, M)$ be a (say real) vector bundle. We can construct in a natural way vector bundles from $E$:

• For any integer $p\geq 1$, we can construct a vector bundle $\bigwedge^p(E)$ over $M$. As a set, $\bigwedge^p(E)=\bigcup\limits_{x\in M}\bigwedge^p(E_x)$, i.e it is the vector bundle whose fiber over a point $x\in M$ is the $p^{th}$ exterior power of $E_x$. Note that while $E$ itself is not a vector space, we still use the same notation $\bigwedge^p$ for it; so you must use context to determine whether we're speaking of $p^{th}$ exterior powers of vector bundles or of vector spaces. Similarly, one can consider $\bigwedge^p(E^*)=\bigcup\limits_{x\in M}\bigwedge^p(E_x^*)$.

• For integers $r,s\geq 0$, we can construct the $E$-tensor bundle $T^r_s(E)$ over $M$. As a set, $T^r_s(E)=\bigcup\limits_{x\in M}T^r_s(E_x)$, so it's the vector bundle whose fiber over $x\in M$ is the vector space $T^r_s(E_x)$. Again, context should tell you whether $T^r_s$ is for vector bundles or for vector spaces.

• By $\Gamma(E)$, we mean the set of smooth maps $\sigma:M\to E$ such that $\pi\circ \sigma=\text{id}_M$. These are called the "smooth sections of the vector bundle $(E,\pi,M)$", i.e the "fields on $M$ with values in $E$". Note that sometimes, people use symbols like $\Gamma^r(E)$ to mean the set of all $C^r$ maps $\sigma:M\to E$ such that $\pi\circ\sigma=\text{id}_M$, and thus $\Gamma^{\infty}(E)$ means the smooth sections (but very often people always want $C^{\infty}$ so they may not specify the extra $\Gamma^{\infty}$).

Therefore, $\Gamma(\bigwedge^p(E))$ means the set of smooth maps $\sigma:M\to \bigwedge^p(E)$ such that for each $x\in M$, $\sigma(x)\in \bigwedge^p(E_x)$. Similarly $\Gamma(T^r_s(E))$ is the set of smooth maps $\sigma:M\to T^r_s(E)$ such that for each $x\in M$, $\sigma(x)\in T^r_s(E_x)$, i.e is an $(r,s)$ tensor on the vector space $E_x$.

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Specializing to the Tangent Bundle

A very important vector bundle is is the tangent bundle to a given manifold: $\pi:TM\to M$, i.e $E=TM$. We can consider the spaces $\bigwedge^p(TM)$ (not so common), $\bigwedge^p(T^*M)$ (more common) and $T^r_s(TM)$. Usually, by abuse/shortenting of notation we write $T^r_s(M)$ instead of $T^r_s(TM)$ (which is the more "correct" and easily generalizable notation). Now,

• A $p$-form on $M$ is by definition a smooth section of the vector bundle $\bigwedge^p(T^*M)$, i.e an element $\omega\in \Gamma(\bigwedge^p(T^*M))$. More explicitly, it is a smooth mapping $\omega:M\to \bigwedge^p(T^*M)$ which assigns to each $x\in M$, an element $\omega(x)\in \bigwedge^p(T_x^*M)$, which by linear algebra is/can be identified with an alternating multilinear functional $\underbrace{T_xM\times\cdots T_xM}_{\text{$p$ times}}\to \Bbb{R}$. Since $\Gamma(\bigwedge^p(T^*M))$ is very long to write, we simply write $\Omega^p(M)$ for this space.

• An $(r,s)$-tensor field on $M$ by definition means a smooth section of the vector bundle $T^r_s(TM)$, i.e an element of $\Gamma(T^r_s(TM))$.

So, being careful with notation, a $p$-form on the cotangent bundle $T^*M$ is $\Omega^p(T^*M):=\Gamma(\bigwedge^p(T^*(T^*M)))$. Note that when we say "p-form on ..." or "tensor field on ..." the "on" is referring to the base manifold. So a tensor field on $TM$ is a more complicated beast than a tensor field on $M$, and a differential form on $T^*M$ is a more complicated beast than a differential form on $M$ etc. Since the notation can immediately get very out of hand I prefer to say things out in words (it's usually clearer, easier to interpret and much less cumbersome).

For example, in a Riemannian manifold $(M,g)$, the object $g$ is a $(0,2)$-tensor field on $M$, i.e $g\in \Gamma(T^0_2(TM))$ (which happens to be symmetric and non-degenerate). Next, in Hamiltonian mechanics, given a "configuration manifold" $Q$, the cotangent bundle $T^*Q$ has a natural symplectic structure, and the tautological form $\theta$ is a 1-form on $T^*Q$, i.e $\theta\in \Gamma(T^*(T^*Q))$, and the symplectic form is $\omega:=d\theta$ is a $2$-form on $T^*Q$, so $\omega\in \Gamma(\bigwedge^2(T^*(T^*Q)))$.

Answer 2

The tangent and cotangent bundles are examples of vector bundles, which you can think of as families of vector spaces $E=\{V_x\}$, parametrized by the points $x$ of a manifold $M$ (in this case, the (co)tangent spaces). A section of such a bundle $E$ is a (smooth) map $s:M\rightarrow E$ that associates to each point $x\in M$ a vector in $V_x$. For instance, a vector field is a section of $TM$ and a $1$-form a section of $T^*M$. For any bundle $E$, the space of this sections is denoted $\Gamma(E)$.

The usual operations of linear algebra (tensor products, taking the dual etc.) admit a natural extension to these vector bundles, by just applying the operation at each point of the manifold. In particular, the operations of taking the space of tensors $T^{n,m}V=V^{\otimes n}\otimes (V^*)^{\otimes m}$ and the exterior algebra of the dual $\bigwedge^kV^*$ (the space of antisymmetric $(0,k)$-tensors) can be extended to vector bundles.

Now the space of $(n,m)$-tensor fields is just: $$\mathcal{T}^{n,m}(M)=\Gamma(T^{n,m}TM) $$ and similarly that of $k$-forms is: $$\Omega^k(M)=\Gamma(\bigwedge^kTM^*) $$