What might be the definition of a positively oriented chart in From Calculus to Cohomology?

Question

My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave.

I recently finished most of An Introduction to Manifolds by Loring W. Tu, so based on the preface of From Calculus to Cohomology, I started at Chapter 8. I don't believe I've missed anything since charts are first introduced in Chapter 8.

Question: What's a positively oriented chart, first mentioned in Proposition 10.2, please?

---

Some context:

I think this is relevant in answering my other question:

Why is there a form with compact support on a connected oriented manifold with integral one but with support contained in a given open proper subset?

• I think I have to prove either the chart $(U, g: U \to g(U) = U')$ or some restriction $(W, g|_W:W \to g(W))$, $W$ open in $U$, is a "positively oriented chart" or at least an "oriented chart" in order to apply Proposition 10.2

---

My guesses:

1. The definition of "oriented chart" in the book (see also previous definitions of orientation) is meant to be "positively oriented chart" with "negatively oriented chart" to be for orientation-reversing. I mean that

   • 1.1. a chart $(U,h:U \to U')$ is an oriented chart if and only if it is a member of an oriented atlas of an oriented smooth $n$-dimensional manifold, and we sometimes omit $U$ and $U'$ and call $h$, the coordinate map, an oriented chart (instead of something like "oriented map")

   • 1.2 An oriented chart $(U,h:U \to U')$, or just $h$, is positively oriented

     • if and only if $h:U \to U'$ is an orientation-preserving diffeomorphism
     • if and only if $\det(D_q(h)) > 0$
     • if and only if $D_qh: T_qU = T_qM \to T_{h(q)}U' = T_{h(q)} \mathbb R^n$ is an orientation-preserving diffeomorphism of manifolds (See here and here)
     • if and only if $D_qh: T_qU = T_qM \to T_{h(q)}U' = T_{h(q)} \mathbb R^n$ is an orientation-preserving vector space isomorphism of tangent spaces

2. In Proposition 10.2, what is meant by "positively oriented chart" is simply "oriented chart" if we go with the convention that "oriented charts" are "positively oriented charts", as originally in the book.

---

I also tried looking up other books:

1. An Introduction to Manifolds by Loring W. Tu:

   Based on Section 21.5 and Subsection 23.4, I believe the definition for integration is for a chart in an "oriented atlas" of $M$, where an "oriented atlas" is defined one where overlapping charts have positive Jacobian determinant. Thus, "oriented atlas" in An Introduction to Manifolds seems to be the same as "positive atlas" in From Calculus to Cohomology.

2. Manifolds, Tensor Analysis, and Applications by Ralph Abraham, Jerrold E. Marsden, Tudor Ratiu:

   It seems a coordinate chart is defined as positively oriented if the coordinate chart's coordinate map has all its differentials to be orientation preserving (as in vector spaces or as in manifolds, if we still have such equivalence of the 2 notions of orientation preserving).

   • If this is what is meant, then to clarify, do we, once again, have a notion, namely the notion of positively oriented chart, that is actually rooted in some prerequisite algebra notion?

   • I'm not sure this is (exactly) what Madsen and Tornehave mean because there is a difference in definition for manifolds.

   • Update: Based on the proof of Theorem 11.9, which relies on Lemma 11.8, I think this might be the definition or at least equivalent to, implied by or implies the definition.

3. Introduction to Smooth Manifolds by John M. Lee:

   It seems the definition is that for an oriented smooth $n$-manifold $M$ with or without boundary, for a coordinate chart $(U,\varphi) = (U,x^1,...,x^n)$ in the differentiable structure of $M$ (see Tu Subsection 5.3), where $x^i=r^i \circ \varphi$, where $r^1, ..., r^n$ are the standard coordinates on $\mathbb R^n$, $(U,\varphi)$ is said to be positively oriented if the frame $\{\frac{\partial}{\partial x^1}, ..., \frac{\partial}{\partial x^n}\}$ is positively oriented. I think there's no explicit concept of "manifold with boundary" or "frame" in From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave so far, and so if we were to adopt this definition,

"if the frame $\{\frac{\partial}{\partial x^1}, ..., \frac{\partial}{\partial x^n}\}$ is positively oriented"

would be translated to

"if each element of the set $\{\frac{\partial}{\partial x^1}|_p, ..., \frac{\partial}{\partial x^n}|_p\}_{p \in M}$ is positively oriented".

Since each element is a basis of the tangent space $T_pM$, based on Tu Subsection 21.3 (Tu says it was in Subsection 12.5, but I'm not sure that was explicit unless Subsection 12.5 was understood in the context of Proposition 8.9), and this is indeed defined after Definition 9.8

Answer 1

A very long question!

As you know, the concept of orientation arises in linear algebra by taking equivalence classes of ordered bases of a real vector space $V$, two such bases $\{b_i \}$ and $\{b'_i \}$ being equivalent if the linear automorphism sending $b_i$ to $b'_i$ has positive determinant. There are exactly two orientations of a vector space $V$ with dimension $> 0$. For a general $V$ none of these two orientations is privileged and it would be an arbitrary choice to call one of them positive and the other negative. However, if $\omega$ is an orientations of $V$, it makes sense to write $-\omega$ for the other orientation, i.e. the minus-sign indicates that the orientation is reversed. Note that a linear isomorphism $f : V \to W$ between vectors spaces $V,W$ establishes a bijection between ordered bases of $V,W$, and thus between orientations of $V,W$. We can therefore say that linear isomorphisms transfer orientations between vectors spaces.

In contrast to the general case, $\mathbb{R}^n$ as the standard model of an $n$-dimensional real vector space has a canonical ordered basis $\{ e_1,\dots,e_n \}$, and its equivalence class is customarily denoted as the positive orientation of $\mathbb{R}^n$. This special situation is due to the fact that the set $\{ 1,\dots,n \}$ has a natural order.

There are various equivalent approaches to define the concept of an orientation on a differentiable manifold $M$. In my opinion the best approach is to define an orientation of $M$ to be a family $\Omega = (\omega_p)_{p \in M}$ of compatible orientations of the tangent spaces $T_pM$. But what is meaning of compatible? The problem is that $T_{p_i}M$ are distinct for $p_1 \ne p_2$, thus we cannot say that the orientations $\omega_{p_i}$ of $T_{p_i}M$ agree.

Let us first consider the simple case of an open subset $V \subset \mathbb{R}^n$. The tangent spaces $T_xV$, $x \in V$, are all distinct, but there is a canonical linear isomorphism $h_x : T_xV \to \mathbb{R}^n$. This allows to define an orientation of $V$ to be a family of orientations $(\omega_x)_{x \in V}$ of orientations of $T_xV$ such each each $x_0 \in V$ has an open neigborhood $V_{x_0} \subset V$ such that for each $x \in V_{x_0}$, $h_x$ transfers $\omega_x$ to the same orientation of $\mathbb{R}^n$. It is easy to see that a connected $V$ has exactly two orientations. We can moreover say that an orientation of $V$ is positive if each $h_x$ transfers $\omega_x$ to the positive orientation of $\mathbb{R}^n$. Finally, if $R : \mathbb{R}^n \to \mathbb{R}^n$ is a reflection at a hyperplane, e.g. $R(x_1,\dots,x_n) = (-x_1,x_2\dots, x_n)$, then we see that the diffeomorphism $R_V = R : V \to R(V)$ has the property $h_{R(x)} \circ T_xR_V = -h_x$, i.e. $R_V$ is orientation reversing.

An orientation of a differentiable manifold $M$ is now defined as a family of orientations $\Omega = (\omega_p)_{p \in M}$ of $T_pM$ such that for each chart $\phi : U \to V \subset \mathbb{R}^n$ the family $\phi_*(\Omega) = (T_{\phi^{-1}(x)}\phi(\omega_{\phi^{-1}(x)})_{x \in V})$ is an orientation of $V$. The chart $\phi$ is said to be positively (negatively) oriented with respect to $\Omega$ if $\phi_*(\Omega)$ is the positive (negative) orientation of $V$. Obviously each chart on a connected $U$ is either positively or negatively oriented. If $U$ is not connected, we can only say that the restriction $\phi_\alpha$ of $\phi$ to each component $U_\alpha$ of $U$ is either positively or negatively oriented. Moreover, for each chart $\phi : U \to V$ there exists a chart $\phi' : U \to V'$ such that $h_{\phi'(p)}(\phi'_*(\omega_p)) = - h_{\phi(p)}(\phi_*(\omega_p))$ for all $p \in U$ (simply take a reflection $R : \mathbb{R}^n \to \mathbb{R}^n$ and define $\phi' = R_V \circ \phi : U \to V' = R(V)$). Working componentwise, we see that on each chart domain (which is an open subset $U \subset M$ which occurs as the domain of a chart) there exist both positively and negatively oriented charts.

The collection of all positively oriented charts forms an atlas for $M$. All transition functions between charts in this atlas have the property that the sign of the determinant of the Jacobian matrix is $+1$ at each point. Note that the collection of all negatively oriented charts has the same property.

Any atlas having the above property it called an orientable atlas, and this is an alternative way to introduce the concept of orientation on manifolds.

Note, however, that there are no open subsets $U \subset M$ which are positively oriented in an absolute sense: Positive orientation is a property of charts with respect to an orientation $\Omega$.