Manifolds and Tangent Spaces

Manifolds

Definition 1 (Manifold). Let $d < n \in \N$. We say a non-empty set $M \subseteq \R^n$ is a $d$-dimensional $C^1$-manifold if for every $x \in M$ there exists $U \subseteq \R^n$, a reordering of coordinates, and a $C^1$ function of $d$ variables, $g$, such that in the reordered coordinates $U \cap M$ is the graph of $g$. More precisely, for every $x \in M$, there exists a permutation matrix $P$, $U \subseteq \R^n$ open, $U’ \subseteq \R^d$ open, and a function $g \colon U’ \to \R^{n-d}$ such that \begin{equation} U \cap M = \set{ P(x’, g’(x’)) \st x’ \in U’} \end{equation}

Remark 2. If $d = n$, we say $M \subseteq \R^d$ is a $d$-dimensional manifold if $M$ is non-empty and open. (You can think of this as a degenerate case of Definition 1 when $d = n$.)

Remark 3. A connected $1$-dimensional manifold is called a curve and a $2$-dimensional manifold is called a surface.

Proposition 4. If $f \colon \R^n \to \R^m$ is $C^1$, and $c \in \R^m$ is such that $\rank(Df_x) = m$ for every $x \in \set{f = c}$, then $\set{f = c}$ is a $n - m$ dimensional $C^1$-manifold.

Example 5. The unit sphere $S^d \defeq \set{x \in \R^{d+1} \st \abs{x} = 1}$ is a $d$-dimensional manifold in $\R^{d+1}$.

Proposition 6. Let $U \subseteq \R^d$ be open, $d < n$, and $\varphi \colon U \to \R^n$ be a $C^1$ injective function. Suppose for every $x \in U$, $\rank(D\varphi_x) = d$, and there exists $\delta_0 > 0$ such that for every $\delta \in (0, \delta_0)$ there exists $\epsilon > 0$ such that $\varphi^{-1}(B(\varphi(x), \epsilon)) \subseteq B(x, \delta)$.

  1. The image $\varphi(U) \subseteq \R^n$ is a $d$-dimensional $C^1$ manifold.
  2. For every $a \in U$, there exists an open set $V \subseteq \R^n$ and a $C^1$ function $\psi \colon V \to \R^d$ such that $\varphi(a) \in V$, and for every $x \in \varphi^{-1}(V)$, we have $\psi \circ \varphi(x) = x$.

Remark 7. The $\epsilon$-$\delta$ condition above is to avoid situations where the image could loop back on itself in a bad way. For instance, let $U = (0, 2) \subseteq \R$ and consider the function $\varphi \colon U \to \R^2$ defined by \begin{equation} \varphi(t) = \begin{cases} (\cos(2\pi t), \sin(2\pi t)) & 0 < t < 1\\ (t, 0) & 1 \leq t < 2 \,. \end{cases} \end{equation} Note $\varphi(1) = (1, 0)$, but $\varphi(U)$ has three branches meeting at this point and can’t be a manifold.

For completeness, we also provide the more standard / modern “intrinsic” definition of manifold.

Definition 8. Let $M$ be a metric space. We say $M$ is a $d$-dimensional manifold if for every $x\in M$ there exists an open set $U_x \subseteq M$ and a homeomorphism $\varphi_x \colon U_x \to B(0, 1) \subseteq \R^d$.

We say $M$ is a $d$-dimensional $C^1$-manifold if further, for any $x, y \in M$ such that $U_x \cap U_y \neq \emptyset$, the function $\varphi_x \circ \varphi_y^{-1}$ is a $C^1$ function on the set $\varphi_y(U_x \cap U_y) \subseteq \R^d$.

Remark 9. The functions $\varphi_x$ above are called a charts, and the collection $\set{(\varphi_x, U_x)}$ is called an atlas. The functions $\varphi_x^{-1} \circ \varphi_y \colon U_x \cap U_y \to U_x$ is called a transition map / coordinate change function.

Proposition 10. If $M \subseteq \R^n$ is a $d$-dimensional $C^1$ manifold according to Definition 1 then show that $M$ is also a $d$-dimensional $C^1$-manifold according to Definition 8.

Conversely, if $M \subseteq \R^n$ is a $d$-dimensional $C^1$ manifold according to Definition 8 then show that $M$ is also a $d$-dimensional $C^1$-manifold according to Definition 1.

Remark 11. Proposition 10 doesn’t quite show Definitions 1 and 8 are equivalent. Indeed, the converse assumed $M \subseteq \R^n$, which is not required for Definition 8. Showing that a manifold (according to Definition 8) can be embedded into $\R^n$ for some $n$ isn’t easy, and is not easy and is the content of the Nash embedding theorem.

Proof sketch of Proposition 10. For the forward direction, show first that for every $x \in M$ show that there exists an open set $\tilde U_x \ni x$ in $\R^n$, and a bijective function function $\tilde \varphi_x \colon \tilde U_x \to B(0, 1) \subseteq \R^n$ such that $\tilde \varphi_x$ and $\tilde \varphi_x^{-1}$ are both $C^1$, and $\tilde \varphi_x(\tilde U \cap M)$ is contained in a $d$-dimensional subspace of vectors whose last $m = n-d$ coordinates are $0$. You can construct $\tilde \varphi_x$ writing $M$ as the graph $(y, f(y))$ (in reordered coordinates) in a small neighborhood around $x$. Now using the notation $z = (z’, z’’ )$ with $z’ \in \R^d$, and $z’’ \in \R^m$, define $\tilde \varphi_x(z) = (z’ - a’, z’’ - f(z’))$. Now use the inverse function theorem and rescale $\tilde \varphi_x$ if needed.

(The converse is a little trickier, without assuming $\varphi_x^{-1}$ is $C^1$).

Tangent spaces and tangent hyperplanes

Tangent spaces of graphs

Definition 12. Let $U \subseteq \R^d$, $f \colon U \to \R^m$, and $\Gamma = \set{ (x, f(x)) }$ be the graph of $f$. The tangent (hyper)-plane to the graph $\Gamma$ at the point $(a, f(a))$ is \begin{equation} \set{ (x, f(a) + Df_a (x - a)) \st x \in U } \subseteq \R^{d + m}\,. \end{equation} The tangent space at $(a, f(a))$ is defined by \begin{equation} T \Gamma_{(a, f(a))} \defeq \set{ (v, Df_a v) \st v \in \R^d } \subseteq \R^{d + m}\,. \end{equation}

Remark 13. Usually we write points in $\R^d$ as tuples $(x_1, \dots, x_d)$, and convert them to column vectors when needed. That is, $(a, f(a))$ denotes the point $(a_1, \dots, a_d, f_1(a), \dots, f_m(a))$, and the tangent vector $(v, D f_a v)$ denotes \begin{equation} \begin{pmatrix} v_1\\ \vdots\\ v_d\\ Df_a v \end{pmatrix} = \begin{pmatrix} v_1\\ \vdots\\ v_d\\ \grad f_1(a) \cdot v\\ \vdots\\ \grad f_m(a) \cdot v\\ \end{pmatrix} \end{equation}

Remark 14. The tangent plane is the “best linear approximation” to the graph $\Gamma$ at the point $(a, f(a))$.

Example 15. Let $f \colon \R^2 \to \R$, and $\Gamma$ be the graph of $f$. At the point $a$, the tangent space is \begin{equation} x_3 = x_1 \partial_1 f(a) + x_2 \partial_2 f(a)\,. \end{equation} The normal to the tangent space at at a point $(x, f(x))$ is the vector $(\grad f(x), -1)$.

Remark 16. Consider a curve $\Gamma$ in $\R^2$ and $a \in \Gamma$. The usual scenario is that $\Gamma$ “touches” the tangent line at $a$ and the continues (briefly) on the same side of the tangent line. The exception is of course inflection points, where $\Gamma$ passes through its tangent line. In a generic curve, inflection points are usually isolated and this doesn’t happen too often.

In $2D$ however, the picture is quite different. A surface will “touch” and locally stay on the same side of the tangent plane if the Hessian is either positive definite or negative definite. If the Hessian has both a strictly positive and a strictly negative eigenvalue, then the curve will necessarily “pass through” the tangent plane at the point of contact. Further, it is possible to construct surfaces where this happens at every single point. One such example is the surface $z = x^2 - y^2$.

Proposition 17. $T \Gamma_{(a, f(a))}$ is a vector space and $\dim( T\Gamma_{(a, f(a))} ) = d$. The vectors $\set{ (e_i, \partial_i f) \st 1 \leq i \leq d}$ form a basis of $T \Gamma_{(a, f(a))}$.

Proposition 18. The tangent space $T \Gamma_{(a, f(a))}$ is the set of all vectors $v \in \R^n$ such that there exists $r > 0$ and a $C^1$ function $\gamma \colon (-r, r) \to \Gamma$ such that $\gamma(0) = (a, f(a))$ and $\gamma’(0) = v$. [Here $\gamma’(0) = D\gamma_0$, and is sometimes also denoted by $\dot \gamma(0)$.]

Tangent spaces of Manifolds

Definition 19. Let $M \subseteq \R^n$ be a $d$-dimensional manifold. Define $T M_{a}$ is the set of all vectors $v \in \R^n$ such that there exists $r > 0$ and a $C^1$ function $\gamma \colon (-r, r) \to M$ such that $\gamma(0) = a$ and $\gamma’(0) = v$. The tangent plane of $M$ at the point $a$ is defined to be the tangent space shifted to pass through the point $a$.

Remark 20. Since reordering coordinates doesn’t change the dimension of a subspace, Proposition 17 implies $T M_a$ is a subspace of $\R^n$, and $\dim( T M_a ) = d$.

Proposition 21. Let $f \colon \R^n \to \R^m$ be $C^1$, $d = n - m$, $c \in \R^m$ and $M = \set{f = c}$. Suppose for every $x \in M$, $\rank Df_x = m$. Then for every $a \in M$, $T M_a = \ker( Df_a )$, and the tangent plane is given by \begin{equation} \set{x \in \R^n \st Df_a (x - a) = 0} \end{equation}

Proposition 22. If $U \subseteq \R^d$ is open, $\varphi \colon U \to \R^n$ satisfies the conditions of Proposition 6, and $M = \varphi(U)$. Then for any $a \in U$, \begin{equation} T M_{\varphi(a)} = \operatorname{range}(D \varphi_a) = D \varphi_a(\R^d) \,. \end{equation}

Problem 23. Let $f \colon \R^3 \to \R^2$ be $C^1$, $c \in \operatorname{range}(f)$, $S_1 = \set{f_1 = c_1}$, $S_2 = \set{f_2 = c_2}$ and $C = \set{f = c}$. Assume $\rank(Df_x) = 2$ for all $x \in \set{f = c}$. In this case $S_1, S_2$ are surfaces, and $C = S_1 \cap S_2$ is a curve. Let $a \in C$. How does $TS_{i, a}$ relate to $\grad f_i(a)$? How does $TC_a$ relate to $\grad f_1(a) \times \grad f_2(a)$.

Example 24. Let $f(x, y, z) = z^2 - x^2 - y^2 + 1$ and $g(x, y, z) = 2(x - 1) + y - z$ at the point $a = (1, 0, 0)$, and let $S = \set{f = 0}$, $T = \set{g = 0}$, $C = S \cap T$. Find the tangent spaces of $S$, $T$ and $C$ at $a$.

Problem 25. Let $M \subseteq \R^n$ be a $C^1$-manifold, and $U \subseteq \R^n$ be an open set containing $M$. Suppose $f, g \colon U \to \R^m$ are two $C^1$ functions such that $f(x) = g(x)$ for every $x \in M$. Show that for any $a \in M$, $v \in T M_a$, we must have $Df_a v = Dg_a v$.

Problem 26. Let $M, N \subseteq \R^n$ be two $C^1$-manifolds. Let $U \supseteq M$ be open and $\varphi \colon U \to \R^n$ be a $C^1$ function such that $\varphi(M) \subseteq N$. For any $a \in M$, show that $D\varphi_a(TM_a) \subseteq TN_{\varphi(a)}$.

Normal spaces

Definition 27. Let $M \subseteq \R^n$ is a $C^1$ manifold, $a \in M$. The normal space $NM_a$ is defined by \begin{equation} NM_a = T M_a^\perp = \set{u \in \R^n \st u \cdot v = 0 \, \forall v \in T M_a } \,. \end{equation}

Proposition 28. With the same notation as Proposition 21, the vectors $\set{ \grad f_1(a), \dots, \grad f_m(a)}$ form a basis of $N M_a$.

Example 29. When $n = 2$, $m = 1$, $d = 2$, the level set $\set{f = c} \subseteq \R^2$ is a curve, and $\grad f$ is normal to the surface at the point $a$.

Example 30. If $n = 3$, $m = 1$, $d = 2$, the level set $\set{f = c} \subseteq \R^2$ is a surface, and $\grad f$ is normal to the surface at the point $a$.

Remark 31 (Contour plots). One effective way of visualizing surfaces is using contour plots. With this we draw the level sets $\set{f = c}$ for various values of $c$. As long as $Df$ has full rank on the entire level set, we know $\set{f = c}$ will be a curve (but not necessarily connected). Moreover, by Proposition 28, $\grad f$ will be orthogonal to the level curves.

At points where $Df$ does not always have full rank, the level set $\set{f = c}$ may not be a curve. Here is where you get (for instance) some points where the level set looks like the center of a figure eight.

Remark 32. For typical functions (more precisely, for Morse functions), if $\set{f = c_1}$ and $\set{f = c_2}$ don’t have the same topology, then $f$ must have a critical point in the region $\set{c_1 \leq f \leq c_2}$. (This is a generalization of the Mountain pass theorem.)

Problem 33. Let $U \subseteq \R^d$, $f \colon U \to \R$ and $\Gamma$ be the graph of $f$. Express $N \Gamma_{(a, f(a))}$ in terms of $\grad f$.

Hint. Check \begin{equation} N \Gamma_{(a, f(a))} = \Span\set{ (-\grad f(a), 1) } = \Span\Set{ \begin{pmatrix} -\grad f(a)\\ 1 \end{pmatrix} } \,. \end{equation}

Problem 34. For each of the following implicitly defined sets $S$, decide if there is an open neighborhood $U$ of the given point such that $S \cap U$ is a manifold. If yes, find a basis of the tangent space, and as many many linearly independent normal vectors as possible at the given point. Also find the tangent line (or tangent plane).

  1. $x \sin(x) = y + x e^y$ at $(0, 0)$.
  2. $\ln(xy) = y - x$ at $(1, 1)$
  3. $x\sin y + y \sin z + z \sin x = 0$ at $(0, \pi, 2\pi)$.
  4. $z^2 = x^2 + y^2 - 1$ and $2(x - 1) + y - z = 0$ at $(1, 0, 0)$.

Problem 35. Given $U \subseteq \R^2$, $\varphi \colon U \to \R^3$ satisfying the conditions of Proposition 6, and $a \in U$, let $M = \varphi(U)$. Express $N M_{\varphi(a)}$ in terms of $D \varphi_a$.

Hint. Check $ N M_{\varphi(a)} = \Span\set{ \partial_1 \varphi(a) \times \partial_2 \varphi(a)} $.

Problem 36. Given $U \subseteq \R^2$, $\varphi \colon U \to \R^4$ satisfying the conditions of Proposition 6, and $a \in U$, let $M = \varphi(U)$. Express $N M_{\varphi(a)}$ in terms of $D \varphi_a$.

Problem 37. Let $M, N \subseteq \R^n$ be two manifolds and $a \in M \cap N$. We say $M, N$ intersect transversely at $a$ if $\Span\set{TM_a, TN_a} = \R^n$.

  1. Suppose $f, g \colon \R^2 \to \R$ are $C^1$. Given $c, d \in \R$, let $M = \set{f = c}$, $N = \set{g = d}$ and assume $\grad f$ and $\grad g$ are nonzero on $M$ and $N$ respectively. Let $a \in M \cap N$. Find a necessary and sufficient condition on $\grad f$ and $\grad g$ that guarantees $M$ and $N$ intersect transversely at $a$.
  2. Suppose instead $f \colon \R^3 \to \R^2$, and $g \colon \R^3 \to \R$. As before $c \in \R^2$, and $d \in \R$, let $M = \set{f = c}$, $N = \set{g = d }$ and assume $D f$ and $D g$ have rank $2$ and rank $1$ on $M$ and $N$ respectively. Show that $M, N$ intersect transversely at $a \in M \cap N$ if and only if $(\grad f_1(a) \times \grad f_2(a)) \cdot \grad g(a) \neq 0$.

Orientability of Surfaces

Let $S \subseteq \R^3$ be a surface. For every $x \in S$ we know $\dim(TS_x) = 2$ and $\dim(N S_x) = 1$. We say $\hat n(x)$ is a unit normal to $S$ at the point $x$ if $\hat n(x) \in N S_x$ and $\abs{\hat n(x)} = 1$.

Definition 38. A surface $S \subseteq \R^2$ is called orientable if there exists a continuous choice of a unit normal. That is, there exists a continuous function $\hat n \colon S \to \R^3$ such that for every $x \in S$,  $\abs{\hat n(x)} = 1$, and $\hat n(x) \in N S_x$.

For surfaces that enclose a region one can orient them by choosing the outward (or inward) pointing unit normal.

Problem 39.

  1. Suppose $f \colon \R^2 \to \R$ is $C^1$, and $M = \set{ (x, y, f(x, y)) } \subseteq \R^3$ is the graph of $f$. Show that $M$ is orientable.
  2. Suppose $f \colon \R^3 \to \R$ is $C^1$, $M = \set{f = 0}$ and $\grad f \neq 0$ at all points in $M$. Show that $M$ is orientable.

Example 40. The ellipsoid $x^2 / a^2 + y^2 / b^2 + z^2 / c^2 = 1 \subseteq \R^3$ is orientable.

Problem 41. Let $U = \set{ (r, \theta) \st r \in (-1/2, 1/2),\ \theta \in \R }$. Define $u(\theta) = \cos \theta e_1 + \sin \theta e_2$, and $v(\theta) = \cos (\theta/2) u(\theta) + \sin(\theta/2) e_3$, where $e_i \in \R^3$ are the standard basis vectors. Let $\varphi(r, \theta) = u(\theta) + r v(\theta)$ and let $M = \varphi(U)$.

  1. Show that $M$ is a manifold ($M$ is called a Möbius strip).
  2. Show that $M$ is not orientable.

Hint. Given $a \in M$ and write $a = \varphi(r, \theta) = u + r v$, and find both unit normal vectors at $a$. Now suppose $M$ is orientable, and $\hat n(x)$ gives a continuous choice of a unit normal at every $x \in M$. Consider the path $\gamma(t) = \varphi(0, t)$, and show that $\hat n(\gamma(t)) \cdot e_3 = \pm \cos(t/2)$. Use continuity to argue that you must either always have $\hat n(\gamma(t)) \cdot e_3 = \cos(t/2)$, or always have $\hat n(\gamma(t)) \cdot e_3 = -\cos(t/2)$. In either case show that you get a contradiction because $\gamma(2\pi) = \gamma(0)$.