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The following notes are intended to supplement the material in the text and illustrate how concepts such as smooth surface and the tangent space of a smooth surface can be generalized to \(d\)-dimensional manifolds contained in \(\mathbb{R}^n\).

Manifolds Suppose that \(M \subseteq \mathbb{R}^n\) is nonempty. \(M\) is a \(d\)-dimensional manifold if for every \(\mathbf{p} \in M\) there exist open \(U \subseteq \mathbb{R}^d\) and \(V \subseteq \mathbb{R}^n\) containing \(\mathbf{p}\) and a homeomorphism \(\phi: U \to M \cap V\). Here we recall that \(\phi:U \to V\) is a homeomorphism if \(\phi\) is a bijection (one-to-one and onto) and both \(\phi\) and \(\phi^{-1}\) are continuous. It should be noted that it is necessarily the case that if \(M \subseteq \mathbb{R}^n\) is a \(d\)-dimensional manifold, then \(d \leq n\). Moreover, \(n\)-dimensional manifolds \(M \subseteq \mathbb{R}^n\) are necessarily open.

Charts and atlases The function \(\phi\) in the previous definition is called a chart for \(M\) at the point \(\mathbf{p}\). An atlas for \(M\) is a collection of charts for \(M\) whose ranges cover \(M\). As we will see in a moment, properties of a manifold such as being smooth or orientable are specified by requiring the existence of an atlas with additional properties.

Smooth and regular functions with open domains Suppose that \(U \subseteq \mathbb{R}^m\) is open and \(f:U \to \mathbb{R}^n\). We say that \(f\) is smooth at \(\mathbf{p} \in U\) if all partial derivatives of \(f\) exist at \(\mathbf{p}\) and are continuous. We say \(f\) is smooth if it is smooth at all points of its domain. We say that \(f\) is regular at \(\mathbf{p}\) if it is smooth at \(\mathbf{p}\) and the Jacobian matrix \[ J(f)_\mathbf{p} := \big(\frac{\partial f_i}{\partial x_j} \big|_{\strut \mathbf{x} = \mathbf{p}} \big)_{i,j} \] has full rank. (Recall that an \(m \times n\) matrix has full rank if its rank is \(\min(m,n)\).) If \(f\) is is regular at \(\mathbf{p}\) whenever \(f(\mathbf{p}) = \mathbf{q}\), we say that \(\mathbf{q}\) is a regular value of \(f\). We say that \(f\) is regular if it is regular at every point in its domain.

Smooth Manifolds A \(d\)-dimensional manifold \(M \subseteq \mathbb{R}^n\) is smooth if it has an atlas consisting of regular charts. The following theorem is very useful in establishing that certain sets of points are smooth manifolds.

Theorem: Suppose that \(U \subseteq \mathbb{R}^n\) is open and \(f:U \to \mathbb{R}^m\) is differentiable. If \(\mathbf{q}\) is a regular value of \(f\), then \(M := \{\mathbf{p} \in U \mid f(p) = q\}\) is a smooth manifold of dimension \(n-m\).

Example: If \(a,b,c \ne 0\) and at least one is positive, then \[S := \{(x,y,z) \in \mathbb{R}^3 \mid a x^2 + b y^2 + c z^2 = 1\}\] is a smooth surface. To see this, define \(f(x,y,z) := ax^2 + b y^2 + c z^2\). The Jacobian of \(f\) is \([2ax \ \ 2by \ \ 2cz]\), which is regular unless \(x = y = z = 0\). Since this is not the case if \(f(x,y,z) = 1\), \(S\) is a smooth manifold.

Tangent spaces for Euclidean space If \(\mathbf{p} \in \mathbb{R}^n\), let \(\mathbb{R}^n_{\mathbf{p}}\) denote the set \(\{\mathbf{p}\} \times \mathbb{R}^n\). We will write elements \((\mathbf{p},\mathbf{v})\) as \(\mathbf{v}_{\mathbf{p}}\) and will think of them representing the vector \(\mathbf{v}\) as well as a location for its base point \(\mathbf{p}\). \(\mathbb{R}^n_{\mathbf{p}}\) is a vector space with the operations \(\mathbf{u}_{\mathbf{p}} + \mathbf{v}_{\mathbf{p}} := (\mathbf{u}+\mathbf{v})_{\mathbf{p}}\) and \(a (\mathbf{u}_{\mathbf{p}}) := (a \mathbf{u})_{\mathbf{p}}\). If \(f\) is smooth at \(\mathbf{p} \in U\), define \(Df_{\mathbf{p}} : \mathbb{R}^m_{\mathbf{p}} \to \mathbb{R}^n_{f(\mathbf{p})}\) by \(Df_{\mathbf{p}}(\mathbf{v}_{\mathbf{p}}) := (J(f)_{\mathbf{p}} \mathbf{v})_{f(\mathbf{p})}\).

Tangent spaces for manifolds Given a smooth \(d\)-dimensional manifold \(M \subseteq \mathbb{R}^n\) and a point \(\mathbf{p} \in M\), define \(T_{\mathbf{p}} M\) to be the range of \(D_{\phi^{-1}(\mathbf{p})} \phi\) where \(\phi\) is any regular chart for \(M\) at \(\mathbf{p}\). For this to be a valid definition, we must show that \(T_{\mathbf{p}} M\) does not depend on the choice of regular chart. To see that this, suppose that \(\phi\) and \(\psi\) are charts at \(\mathbf{p}\) and set \(\mathbf{x} := \phi^{-1}(\mathbf{p})\) and \(\mathbf{y} := \psi^{-1}(\mathbf{p})\). The function \(\psi^{-1} \circ \phi\) is a smooth homeomorphism between an open set \(U \subseteq \mathbb{R}^d\) containing \(\mathbf{x}\) and an open set \(V \subseteq \mathbb{R}^d\) containing \(\mathbf{y}\). Thus \(D_{\mathbf{x}} (\psi^{-1} \circ \phi):\mathbb{R}^d_{\mathbf{x}} \to \mathbb{R}^d_{\mathbf{y}}\) is an isomorphism. Since \[ D_{\mathbf{x}} \phi = D_{\mathbf{x}} (\psi \circ (\psi^{-1} \circ \phi)) = (D_{\mathbf{y}} \psi) \circ (D_{\mathbf{x}} (\psi^{-1} \circ \phi)) \] and since \(D_{\mathbf{x}} (\psi^{-1} \circ \phi)\) is onto, the ranges of \(D_{\phi^{-1}(\mathbf{p})} \phi = D_{\mathbf{x}} \phi\) and \(D_{\psi^{-1}(\mathbf{p})} \psi = D_{\mathbf{y}} \psi\) are the same and hence \(T_{\mathbf{p}} M\) is well defined.

Elements of \(T_{\mathbf{p}} M\) are called tangent vectors to \(M\) at \(\mathbf{p}\) and \(T_{\mathbf{p}} M\) is called the tangent space to \(M\) at \(\mathbf{p}\). Define \[ TM := \{\mathbf{v}_{\mathbf{p}} \mid \mathbf{p} \in M \textrm{ and } \mathbf{v}_{\mathbf{p}} \in T_{\mathbf{p}} M\} = \bigcup_{\mathbf{p} \in M} T_{\mathbf{p}} M. \] \(TM \subseteq \mathbb{R}^n \times \mathbb{R}^n\) is called the tangent bundle of \(M\) and itself is a \(2d\)-dimensional smooth manifold. (We will think of \(\mathbb{R}^n \times \mathbb{R}^n\) as being the same as \(\mathbb{R}^{2n}\), even though formally they are different. This way \(TM \subseteq \mathbb{R}^{2n}\) and our definition of "smooth manifold" applies.) A vector field on \(M\) is a closed subset \(\mathbf{F} \subseteq TM\) such that for every \(\mathbf{p} \in M\) there is a unique element \(\mathbf{F}_{\mathbf{p}}\) of \(\mathbf{F} \cap \mathbb{R}^n_{\mathbf{p}}\).

Smooth and regular functions on manifolds Next we would like to define what it means for a function between two manifolds to be smooth. If \(M \subseteq \mathbb{R}^m\) and \(N \subseteq \mathbb{R}^n\), then a function \(f:M \to N\) is smooth at a point \(\mathbf{p} \in M\) if \(\psi^{-1} \circ f \circ \phi\) is smooth at \(\mathbf{x}:=\phi^{-1}(\mathbf{p})\) where \(\phi\) is a smooth chart for \(M\) at \(\mathbf{p}\) and \(\psi\) is a smooth chart for \(N\) at \(f(\mathbf{p})\). Similarly we say that \(f\) is regular at \(\mathbf{p}\) if \(\psi^{-1} \circ f \circ \phi\) is regular at \(\phi^{-1}(\mathbf{p})\).

Differentiation of smooth functions on manifolds. If \(M\) and \(N\) are smooth manifolds and \(f:M \to N\) is a smooth function, we would like to define \(D_{\mathbf{p}} f\) for each \(\mathbf{p} \in M\). Our goal is that this operation should satisfy the following properties:

1. If \(M\) is an open subset of \(\mathbb{R}^n\) for some \(n\), then \(D_{\mathbf{p}} f\) is defined using the Jacobian: \(D_{\mathbf{p}} f (\mathbf{v}_{\mathbf{p}}) := (J(f)_{\mathbf{p}} \mathbf{v})_{f(\mathbf{p})}\).
2. If \(f :M_0 \to M_1\) and \(g:M_1 \to M_2\) are smooth, then \(g \circ f\) is smooth and \(D_{\mathbf{p}} (g \circ f) = (D_{f(\mathbf{p})} g) \circ (D_{\mathbf{p}} f)\).
3. If \(f : M \to M\) is the identity function, then \(D_{\mathbf{p}} f :T_{\mathbf{p}} M \to T_{\mathbf{p}} M\) is the identity transformation.

Notice that properties (2) and (3) would imply that \(D_{f(\mathbf{p})} (f^{-1}) = (D_{\mathbf{p}} f)^{-1}\). Now suppose that \(M \subseteq \mathbb{R}^m\) is a \(d\)-dimensional manifold, \(N\) is a manifold and \(f:M \to N\) is a smooth function and \(\mathbf{p} \in M\). Let \(\phi\) be a regular chart for \(M\) at \(\mathbf{p}\) and let \(\psi\) be a regular chart for \(N\) at \(f(\mathbf{p})\); set \(\mathbf{x}:=\phi^{-1}(\mathbf{p})\) and \(\mathbf{y} = \psi^{-1}(f(\mathbf{p}))\) Observe that \(\psi \circ (\psi^{-1} \circ f \circ \phi) \circ \phi^{-1}\) is defined on the range of \(\phi\) and agrees with \(f\) on its domain. Furthermore, the functions \(\phi\), \(\psi\), and \(\psi^{-1} \circ f \circ \phi\) are all smooth functions which are defined on open subsets of Euclidean space. Thus \(D_{\mathbf{y}} \psi\), \(D_{\mathbf{x}} (\psi^{-1} \circ f \circ \phi)\), and \(D_{\mathbf{y}} \phi\) are all defined using our existing definitions from vector calculus. Anticipating (1), (2) and (3), we define \[ D_{\mathbf{p}} f := \big( D_{\mathbf{y}} \psi \big) \circ \big( D_{\mathbf{x}} (\psi^{-1} \circ f \circ \phi) \big) \circ \big( D_{\mathbf{x}} \phi \big)^{-1}. \] It can be shown that this definition does not depend on the choice of the charts \(\phi\) and \(\psi\). Moreover, this definition of \(D_{\mathbf{p}} f\) satisfies properties (1), (2) and (3) above. Finally, observe that if \(f:M \to N\) is smooth, then the functions \(D_{\mathbf{p}} f:T_{\mathbf{p}} M \to T_{f(\mathbf{p})} N\) are all restrictions of a single continuous function \(Df :T M \to T N\).

Diffeomorphisms and local diffeomorphisms If \(M\) and \(N\) are manifolds a function \(f:M \to N\) is a diffeomorphism if \(f\) is a smooth bijection. We will see later that diffeomorphisms are necessarily regular. Thus if \(f:M \to N\) is a diffeomorphism, then \(Df:TM \to TN\) is a homeomorphism. Next suppose that \(M \subseteq \mathbb{R}^m\) and \(N \subseteq \mathbb{R}^n\) are manifolds. A function \(f:M \to N\) is a local diffeomorphism if for every \(\mathbf{p} \in M\), there is an open set \(U \subseteq \mathbb{R}^m\) such that the restriction \(f \restriction M \cap U\) is a diffeomorphism between \(M \cap U\) and \(N \cap V\) for some open set \(V \subseteq \mathbb{R}^n\). A composition of local diffeomorphisms is again a local diffeomorphism. The following theorems are useful.

Theorem A function \(f:M \to N\) is a local diffeomorphism if and only if \(f\) is smooth and \(D_{\mathbf{p}} f\) is an isomorphism between \(T_{\mathbf{p}} M\) and \(T_{f(\mathbf{p})} N\) for all \(\mathbf{p} \in M\).

Theorem If \(M \subseteq \mathbb{R}^m\) is a \(d\)-dimensional manifold, \(N \subseteq \mathbb{R}^n\) and there is a surjective diffeomorphism \(f:M \to N\), then \(N\) is a \(d\)-dimensional manifold.

Orientable manifolds Suppose that \(M\) is a manifold and \(\mathbf{p} \in M\). We say that charts \(\phi\) and \(\psi\) are compatible at \(\mathbf{p}\) if the \(\det (J(\psi^{-1} \circ \phi)_{\mathbf{x}})\) is positive where \(\mathbf{x}=\phi^{-1}(\mathbf{p})\). Two charts are compatible if they are compatible any point in the intersection of their ranges. A manifold is orientable if it has an atlas which is consists of compatible charts.