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DIFFERENTIAL GEOMETRY COURSE NOTES
KO HONDA
1. REVIEW OF TOPOLOGY AND LINEAR ALGEBRA
1.1. Review of topology.
Definition 1.1. A topological space is a pair (X, T ) consisting of a set X and a collection T = {Uα} of subsets of X, satisfying the following:
(1) ∅, X ∈ T , (2) if Uα, Uβ ∈ T , then Uα ∩ Uβ ∈ T , (3) if Uα ∈ T for all α ∈ I, then ∪α∈I Uα ∈ T. (Here I is an indexing set, and is not necessarily finite.) T is called a topology for X and Uα ∈ T is called an open set of X.
Example 1: Rn^ = R × R × · · · × R (n times) = {(x 1 ,... , xn) | xi ∈ R, i = 1,... , n}, called real n-dimensional space.
How to define a topology T on Rn? We would at least like to include open balls of radius r about y ∈ Rn: Br(y) = {x ∈ Rn^ | |x − y| < r},
where |x − y| =
(x 1 − y 1 )^2 + · · · + (xn − yn)^2.
Question: Is T 0 = {Br(y) | y ∈ Rn, r ∈ (0, ∞)} a valid topology for Rn?
No, so you must add more open sets to T 0 to get a valid topology for Rn.
T = {U | ∀y ∈ U, ∃Br(y) ⊂ U }.
Example 2A: S^1 = {(x, y) ∈ R^2 | x^2 + y^2 = 1}. A reasonable topology on S^1 is the topology induced by the inclusion S^1 ⊂ R^2.
Definition 1.2. Let (X, T ) be a topological space and let f : Y → X. Then the induced topology f −^1 T = {f −^1 (U ) | U ∈ T } is a topology on Y.
Example 2B: Another definition of S^1 is [0, 1]/ ∼, where [0, 1] is the closed interval (with the topology induced from the inclusion [0, 1] → R) and the equivalence relation identifies 0 ∼ 1. A reasonable topology on S^1 is the quotient topology. 1
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Definition 1.3. Let (X, T ) be a topological space, ∼ be an equivalence relation on X, X = X/ ∼ be the set of equivalence classes of X, and π : X → X be the projection map which sends x ∈ X to its equivalence class [x]. Then the quotient topology T of X is the set of V ⊂ X for which π−^1 (V ) is open.
Definition 1.4. A map f : X → Y between topological spaces is continuous if f −^1 (V ) = {x ∈ X|f (x) ∈ V } is open whenever V ⊂ Y is open.
HW: Show that the inclusion S^1 ⊂ R^2 is a continuous map. Show that the quotient map [0, 1] → S^1 = [0, 1]/ ∼ is a continuous map.
More generally,
(1) Given a topological space (X, T ) and a map f : Y → X, the induced topology on Y is the “smallest”^1 topology which makes f continuous. (2) Given a topological space (X, T ) and a surjective map π : X Y , the quotient topology on Y is the “largest” topology which makes π continuous.
When are two topological spaces equivalent? The following gives one notion:
Definition 1.5. A map f : X → Y is a homeomorphism is there exists an inverse f −^1 : Y → X for which f and f −^1 are both continuous.
HW: Show that the two incarnations of S^1 from Examples 2A and 2B are homeomorphic.
Zen of mathematics: Any world (“category”) in mathematics consists of spaces (“objects”) and maps between spaces (“morphisms”).
Examples:
(1) (Topological category) Topological spaces and continuous maps. (2) (Groups) Groups and homomorphisms. (3) (Linear category) Vector spaces and linear transformations.
1.2. Review of linear algebra.
Definition 1.6. A vector space V over a field k = R or C is a set V equipped with two operations V × V → V (called addition) and k × V → V (called scalar multiplication) s.t.
(1) V is an abelian group under addition. (a) (Identity) There is a zero element 0 s.t. 0 + v = v + 0 = v. (b) (Inverse) Given v ∈ V there exists an element w ∈ V s.t. v + w = w + v = 0. (c) (Associativity) (v 1 + v 2 ) + v 3 = v 1 + (v 2 + v 3 ). (d) (Commutativity) v + w = w + v. (2) (a) 1 v = v. (b) (ab)v = a(bv). (c) a(v + w) = av + aw. (^1) Figure out what “smallest” and “largest” mean.
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- REVIEW OF DIFFERENTIATION
2.1. Definitions. Let f : Rm^ → Rn^ be a map. The discussion carries over to f : U → V for open sets U ⊂ Rm^ and V ⊂ Rn.
Definition 2.1. The map f : Rm^ → Rn^ is differentiable at a point x ∈ Rm^ if there exists a linear map L : Rm^ → Rn^ satisfying
(1) lim h→ 0
|f (x + h) − f (x) − L(h)| |h|
where h ∈ Rm^ − { 0 }. L is called the derivative of f at x and is usually written as df (x).
HW: Show that if f : Rm^ → Rn^ is differentiable at x ∈ Rm, then there is a unique L which satisfies Equation (1).
Fact 2.2. If f is differentiable at x, then df (x) : Rm^ → Rn^ is a linear map which satisfies
(2) df (x)(v) = lim t→ 0
f (x + tv) − f (x) t
We say that the directional derivative of f at x in the direction of v exists if the right-hand side of Equation (2) exists. What Fact 2.2 says is that if f is differentiable at x, then the directional derivative of f at x in the direction of v exists and is given by df (x)(v).
2.2. Partial derivatives. Let ej be the usual basis element (0,... , 1 ,... , 0), where 1 is in the jth position. Then df (x)(ej ) is usually called the partial derivative and is written as (^) ∂x∂fj (x) or ∂j f (x).
More explicitly, if we write f = (f 1 ,... , fn)T^ (here T means transpose), where fi : Rm^ → R, then ∂f ∂xj
(x) =
∂f 1 ∂xj
(x),... ,
∂fn ∂xj
(x)
)T
and df (x) can be written in matrix form as follows:
df (x) =
∂f 1 ∂x 1 (x)^...^
∂f 1 ∂xm (x) .. .
∂fn ∂x 1 (x)^...^
∂fn ∂xm (x)
The matrix is usually called the Jacobian matrix.
Facts:
(1) If ∂i(∂j f ) and ∂j (∂if ) are continuous on an open set 3 x, then ∂i(∂j f )(x) = ∂j (∂if )(x). (2) df (x) exists if all ∂f ∂xij (y), i = 1,... , n, j = 1,... , m, exist on an open set 3 x and each ∂fi ∂xj is continuous at^ x.
Shorthand: Assuming f is smooth, we write ∂αf = ∂ 1 α 1 ∂ 2 α 2... ∂α k kf where α = (α 1 ,... , αk).
Definition 2.3.
DIFFERENTIAL GEOMETRY COURSE NOTES 5
(1) f is smooth or of class C∞^ at x ∈ Rm^ if all partial derivatives of all orders exist at x. (2) f is of class Ck^ at x ∈ Rm^ if all partial derivatives up to order k exist on an open set 3 x and are continuous at x.
2.3. The Chain Rule.
Theorem 2.4 (Chain Rule). Let f : R^ → Rm^ be differentiable at x and g : Rm^ → Rn^ be differentiable at f (x). Then g ◦ f : R^ → Rn^ is differentiable at x and
d(g ◦ f )(x) = dg(f (x)) ◦ df (x). Draw a picture of the maps and derivatives.
Definition 2.5. A map f : U → V is a C∞-diffeomorphism if f is a smooth map with a smooth inverse f −^1 : V → U. (C^1 -diffeomorphisms can be defined similarly.)
One consequence of the Chain Rule is:
Proposition 2.6. If f : U → V is a diffeomorphism, then df (x) is an isomorphism for all x ∈ U.
Proof. Let g : V → U be the inverse function. Then g ◦ f = id. Taking derivatives, dg(f (x)) ◦ df (x) = id as linear maps; this give a left inverse for df (x). Similarly, a right inverse exists and hence df (x) is an isomorphism for all x.
DIFFERENTIAL GEOMETRY COURSE NOTES 7
Note: In the rest of the course when we refer to a “manifold”, we mean a “smooth manifold”, unless stated otherwise.
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- EXAMPLES OF SMOOTH MANIFOLDS Today we give some examples of smooth manifolds. For each of the examples, you should also verify the Hausdorff and second countable conditions!
(1) Rn^ is a smooth manifold.^3 Atlas: {id : U = Rn^ → Rn} consisting of only one chart.
(2) Any open subset U of a smooth manifold M is a smooth manifold. Given an atlas {φα : Uα → Rn}) for M , an atlas for U is {φα|U ∩Uα : Uα ∩ U → Rn}.
(3) Let Mn(R) be the space of n × n matrices with real entries, and let
GL(n, R) = {A ∈ Mn(R) | det(A) 6 = 0}.
GL(n, R) is an open subset of Mn(R) ' Rn 2 , hence is a smooth n^2 -dimensional manifold. GL(n, R) is called the general linear group of n × n real matrices.
(4) If M and N are smooth m- and n-dimensional manifolds, then their product M × N can naturally be given the structure of a smooth (m + n)-dimensional manifold. Atlas: {φα × ψβ : Uα × Vβ → Rm^ × Rn}, where {φα : Uα → Rm} is an atlas for M and {ψβ : Vβ → Rn} is an atlas for N.
(5) S^1 = {x^2 + y^2 = 1} is a smooth 1-dimensional manifold.
(i) One possible atlas: Open sets U 1 = {y > 0 }, U 2 = {y < 0 }, U 3 = {x > 0 }, U 4 = {x < 0 }, together with projections to the x-axis or the y-axis, as appropriate. Check the transition maps! (ii) Another atlas: Open sets U 1 = {y 6 = 1} and U 2 = {y 6 = − 1 }, together with stereographic projections from U 1 to y = − 1 and U 2 to y = 1. The map φ 1 : U 1 → R is defined as follows: Take the line L(x,y) which passes through (0, 1) and (x, y) ∈ U 1. Then let φ 1 be the x-coordinate of the intersection point between L(x,y) and y = − 1. The map φ 2 : U 2 → R is defined similarly by projecting from (0, −1) to y = 1. Check the transition maps!
(6) Sn^ = {x^21 + · · · + x^2 n+1 = 1} ⊂ Rn+1. Generalize the discussion from (5).
(7) In dimension 2, S^2 , T 2 , genus g surface.
(8) (Real projective space) RPn^ = (Rn+1^ − {(0,... , 0)})/ ∼, where
(x 0 , x 1 ,... , xn) ∼ (tx 0 , tx 1 ,... , txn), t ∈ R − { 0 }.
RPn^ is called the real projective space of dimension n. The equivalence class of (x 0 ,... , xn) is denoted by [x 0 ,... , xn].
(^3) Strictly speaking, this should say “can be given the structure of a smooth manifold”. There may be more than one
choice and we have not yet discussed when two manifolds are the same.
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- SMOOTH FUNCTIONS AND SMOOTH MAPS Today we discuss smooth functions on a manifold and smooth maps between manifolds.
5.1. Choice of atlas. Let (M, T ) be the underlying topological space of a manifold, and A 1 = {(Uα, φα)}, A 2 = {(Vβ , ψβ )} be two atlases.
Question: When do they represent the same smooth manifold?
Definition 5.1. Two atlases A 1 and A 2 on M are compatible if
φα(Uα ∩ Vβ )
ψβ ◦φ− α^1 −→ ψβ (Uα ∩ Vβ )
is a smooth map for all pairs Uα ∩ Vβ 6 = ∅.
If A 1 and A 2 are compatible, then we can take A = A 1 ∪ A 2 which is compatible with both A 1 and A 2.
Definition 5.2. Given a smooth manifold (M, A), its maximal atlas Amax = {(Uα, φα)} is an atlas which is compatible with A and contains every atlas A′^ ⊃ A which is compatible with A.
5.2. Smooth functions.
Some more zen: You can study an object (such as a manifold) either by looking at the object itself or by looking at the space of functions on the object. In the topological category, the space of functions would be C^0 (M ), the space of continuous functions f : M → R. The function space perspective has been especially fruitful in algebraic geometry.
Question: What is the appropriate space of functions for a smooth manifold (M, A)?
Definition 5.3. Given a smooth manifold (M, A), a function f : M → R is smooth if
f ◦ φ− α 1 : φα(Uα) → R
is smooth for each coordinate chart (Uα, φα) of A.
Note that the definition of a smooth function on M depends on the atlas A.
The space of smooth functions f : M → R with respect to A is written as CA∞ (M ). When A is understood, we write C∞(M ).
Lemma 5.4. Two atlases A 1 and A 2 are compatible if and only if CA∞ 1 (M ) = CA∞ 2 (M ).
Proof. Suppose A 1 = {(Uα, φα)} and A 2 = {(Vβ , ψβ )} are compatible. It suffices to show that CA∞ 1 (M ) ⊃ CA∞ 2 (M ). If f ∈ C∞A 2 (M ), then f ◦ ψ β− 1 : ψβ (Vβ ) → R is smooth for all β. Now
(3) f ◦ φ− α 1 : φα(Uα ∩ Vβ ) → R
can be written as (f ◦ ψ− β 1 ) ◦ (ψβ ◦ φ− α 1 ), and each of f ◦ ψ− β 1 and ψβ ◦ φ− α 1 is smooth (the latter is smooth because A 1 and A 2 are compatible); hence (3) is smooth for all α and β. This implies that f ◦ φ− α 1 : φα(Uα) → R is smooth for all α.
DIFFERENTIAL GEOMETRY COURSE NOTES 11
Suppose CA∞ 1 (M ) = C∞A 2 (M ). We use the existence of bump functions, i.e., smooth functions h : R → [0, 1] such that h(x) = 1 on [a, b] and h(x) = 0 on R − [c, d], where c < a < b < d. (The construction of bump functions is an exercise.) In order to show that the transition maps ψβ ◦ φ− α 1 : φα(Uα ∩ Vβ ) → ψβ (Uα ∩ Vβ ) ⊂ Rn
are smooth, we postcompose with the projection πj : Rn^ → R to the jth R factor and show that πj ◦ ψβ ◦ φ− α 1 is smooth. Given x ∈ ψβ (Uα ∩ Vβ ), let Bε(x) ⊂ B 2 ε(x) ⊂ ψβ (Uα ∩ Vβ ) be small open balls around x. Using the bump functions we can construct a function f on Uα ∩ Vβ such that f ◦ ψ− β 1 equals πj on Bε(x) and 0 outside B 2 ε(x); f can be extended to the rest of M by setting
f = 0. f is clearly in CA∞ 2 (M ). Since CA∞ 1 (M ) = CA∞ 2 (M ), f ◦ φ− α 1 = (f ◦ ψ β− 1 ) ◦ (ψβ ◦ φ− α 1 ) is smooth. This is sufficient to show the smoothness of πj ◦ ψβ ◦ φ− α 1 and hence of ψβ ◦ φ− α 1.
Pullback: Let φ : X → Y be a continuous map between topological spaces. Then there is a naturally defined pullback map φ∗^ : C^0 (Y ) → C^0 (X)
given by f 7 → f ◦ φ. Note that pullback is contravariant, i.e., the direction is from Y to X, which is the opposite from the original map φ.
Consider the smooth manifold (M, A). If ψ : M → M is a homeomorphism, then ψ∗^ : C^0 (M ) →∼ C^0 (M ). Although CA∞ (M ) ∼ → ψ∗(CA∞ (M )), in general C∞A (M ) 6 = ψ∗(CA∞ (M )).
Definition 5.5. Two C∞-structures CA∞ 1 (M ) and C∞A 2 (M ) are equivalent if there exists a homeo- morphism of M which takes CA∞ 1 (M ) ' CA∞ 2 (M ).
Amazing fact: (Milnor) S^7 has several inequivalent smooth structures! (Not amazingly, S^1 has only one smooth structure.)
Major open question: (Smooth Poincar´e Conjecture) How many smooth structures does S^4 have?
5.3. Smooth maps. In the category of smooth manifolds, we need to define the appropriate maps, called smooth maps.
Definition 5.6. A map φ : M → N between manifolds is smooth if for any p ∈ M there exist coordinate charts (Uα, φα), (Vβ , ψβ ) such that Uα 3 p, Vβ 3 f (p), and the composition
φα(Uα) φ− α^1 → Uα φ → Vβ
ψβ → ψβ (Vβ )
is smooth.
Remark 5.7. For the above definition, we need to take Uα 3 p which is “sufficiently small” so that φ(Uα) ⊂ Vβ. So this means that we should be using a maximal atlas (or at least a “large enough” atlas).
Lemma 5.8. φ : M → N is smooth if and only if φ∗(C∞(N )) ⊂ C∞(M ).
DIFFERENTIAL GEOMETRY COURSE NOTES 13
- THE INVERSE FUNCTION THEOREM
6.1. Inverse function theorem.
Definition 6.1. A smooth map f : M → N between two manifolds is a diffeomorphism if there is a smooth inverse f −^1 : N → M.
The inverse function theorem, given below, is the most important basic theorem in differential geometry. It says that an isomorphism in the linear category implies a local diffeomorphism in the differentiable category. Hence we can move from “infinitesimal” to “local”.
Theorem 6.2 (Inverse function theorem). Let f : U → V be a C^1 map, where U and V are open sets of Rn. If df (x) : Rn^ → Rn^ is an isomorphism, then f is a local diffeomorphism near x, i.e., there exist open sets Ux 3 x and Vf (x) 3 f (x) such that f |Ux : Ux → Vf (x) is a diffeomorphism.
Partial proof. Refer to Spivak, Calculus on Manifolds for a complete proof. Assume without loss of generality that x = 0 and f (0) = 0. We will only show that for all y ∈ V near 0 there exists x′^ ∈ U near 0 such that f (x′) = y. First pick x 1 such that df (0)(x 1 ) = y; this is possible since df (0) is an isomorphism. We then compare f (x 1 ) and df (0)(x 1 ) = y: By the differentiability of f , for any sufficiently small ε > 0 there exists δ > 0 such that whenever |x 1 | < δ we have:
|f (x 1 ) − f (0) − df (0)(x 1 )| = |f (x 1 ) − y| ≤ ε|x 1 |.
In other words, the error |f (x 1 ) − y| is much smaller than |x 1 |. Next we take x 2 such that df (x 1 )(x 2 ) = y − f (x 1 ). Then we have:
|f (x 1 + x 2 ) − f (x 1 ) − df (x 1 )(x 2 )| = |f (x 1 + x 2 ) − y| ≤ ε|x 2 |.
Now, since f is in the class C^1 , df (˜x) is invertible for all x˜ near 0 and there exists a constant C > 0 such that the norm of (df (˜x))−^1 is < C. Hence
|x 2 | < C|df (x 1 )(x 2 )| = C|y − f (x 1 )| ≤ Cε|x 1 |.
We then repeat the process to obtain x 1 , x 2 ,... , and f (x 1 + x 2 +... ) = y. (This process is usually called Newton iteration.)
6.2. Illustrative example. Let f : R^2 → R, (x, y) 7 → x^2 + y^2. We would like to analyze the level sets f −^1 (a), where a > 0. To that end, we consider
F : R^2 → R^2 , (x, y) 7 → (f (x, y), y).
Let us use coordinates (x, y) for the domain R^2 and coordinates (u, v) for the range R^2. We compute:
dF (x, y) =
2 x 2 y 0 1
Let us restrict our attention to the portion x > 0. Since det(dF (x, y)) = 2x > 0 , the inverse function theorem applies and there is a local diffeomorphism between a neighborhood U(x,y) ⊂ R^2
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of a point (x, y) on the level set f (x, y) = a and a neighborhood VF (x,y) of F (x, y) on the line u = a.
In particular, f −^1 (a)∩U(x,y) is mapped to {u = a}∩VF (x,y); in other words, F is a local diffeomor- phism which “straightens out” f −^1 (a). Hence f −^1 (a), restricted to x > 0 , is a smooth manifold. Check the transition functions!
Interpreted slightly differently, the pair f, y can locally be used as coordinate functions on R^2 , provided x > 0.
6.3. Rank. Recall that the dimension of a vector space V is the cardinality of a basis for V. If V is finite-dimensional, then V ' Rm^ for some m, and dim V = m.
Definition 6.3. The rank of a linear map L : V → W is the dimension of im(L).
Definition 6.4. The rank of a smooth map f : Rm^ → Rn^ at x ∈ Rm^ is the rank of df (x) : Rm^ → Rn. The map f has constant rank if the rank of df (x) is constant.
We can similarly define the rank of a smooth map f : M → N at a point x ∈ M by using local coordinates.
Claim 6.5. The rank at x ∈ M is constant under change of coordinates.
Proof. We compare the ranks of d(ψα ◦ f ◦ φ− α 1 ) and d(ψβ ◦ f ◦ φ− β 1 ), where φα : Uα → Rm, ψα : Vα → Rn, Uα ⊂ M , Vβ ⊂ N , and φβ , ψβ are defined similarly. The invariance of rank is due to the chain rule:
d(ψβ ◦ f ◦ φ− β 1 ) = d((ψβ ◦ ψ− α 1 ) ◦ (ψα ◦ f ◦ φ− α 1 ) ◦ (φ− α 1 ◦ φβ )) = d(ψβ ◦ ψ− α 1 ) ◦ d(ψα ◦ f ◦ φ− α 1 ) ◦ d(φ− α 1 ◦ φβ ),
and by observing that d(ψβ ◦ ψ− α 1 ) and d(φ− α 1 ◦ φβ ) are linear isomorphisms.
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The Jacobian is df (x, y) = (2x, 2 y). Since x and y are never simultaneously zero, the rank of df is 1 at all points of R^2 − {(0, 0)} and in particular on S^1. Using the implicit function theorem, it follows that S^1 is a manifold.
7.2. Regular values and Sard’s theorem.
Definition 7.5. Let f : M → N be a smooth map.
(1) A point y ∈ N is a regular value of f if df (x) is surjective for all x ∈ f −^1 (y). (2) A point y ∈ N is a critical value of f if df (x) is not surjective for some x ∈ f −^1 (y). (3) A point x ∈ M is a critical point of f if df (x) is not surjective. The implicit function theorem implies that f −^1 (y) can be given the structure of a manifold if y is a regular value of f.
Example: Let M = {x^3 + y^3 + z^3 = 1} ⊂ R^3. Consider the map
f : R^3 → R, (x, y, z) 7 → x^3 + y^3 + z^3.
Then M = f −^1 (1). The Jacobian is given by df (x, y, z) = (3x^2 , 3 y^2 , 3 z^2 ) and the rank of df (x, y, z) is one if and only if (x, y, z) 6 = (0, 0 , 0). Since (0, 0 , 0) 6 ∈ M , it follows that 1 is a regular value of f. Hence M can be given the structure of a manifold.
HW: Prove that Sn^ ⊂ Rn+1^ is a manifold.
Example: Zero sets of homogeneous polynomials in RPn. A polynomial f : Rn+1^ → R is homogeneous of degree d if f (tx) = tdf (x) for all t ∈ R − { 0 } and x ∈ Rn+1. The zero set Z(f ) of f is given by {[x 0 ,... , xn] | f (x 0 ,... , xn) = 0}. By the homogeneous condition, Z(f ) is well-defined. We can check whether Z(f ) is a manifold by passing to local coordinates. For example, consider the homogeneous polynomial f (x 0 , x 1 , x 2 ) = x^30 + x^31 + x^32 of degree 3 on RP^2. Consider the open set U = {x 0 6 = 0} ⊂ RP^2. If we let x 0 = 1, then on U ' R^2 we have f (x 1 , x 2 ) = 1 + x^31 + x^32. Check that 0 is a regular value of f (x 1 , x 2 )! The open sets {x 1 6 = 0} and {x 2 6 = 0} can be treated similarly.
More involved example: Let SL(n, R) = {A ∈ Mn(R) | det(A) = 1}. SL(n, R) is called the special linear group of n × n real matrices. Consider the determinant map
f : Rn
2 → R, A 7 → det(A).
We can rewrite f as follows:
f : Rn^ × · · · × Rn^ → R, (a 1 ,... , an) 7 → det(a 1 ,... , an),
where ai are column vectors and A = (a 1 ,... , an) = (aij ).
First we need some properties of the determinant:
(1) f (e 1 ,... , en) = 1. (2) f (a 1 ,... , ciai + c′ ia′ i,... , an) = ci · f (a 1 ,... , ai,... , an) + c′ i · f (a 1 ,... , a′ i,... , an). (3) f (... , ai, ai+1,... ) = −f (... , ai+1, ai,... ).
DIFFERENTIAL GEOMETRY COURSE NOTES 17
(1) is a normalization, (2) is called multilinearity, and (3) is called the alternating property. It turns out that (1), (2), and (3) uniquely determine the determinant function.
We now compute df (A)(B):
df (A)(B) = lim t→ 0
f (A + tB) − f (A) t = lim t→ 0
det(a 1 + tb 1 ,... , an + tbn) − det(a 1 ,... , an) t = lim t→ 0
det(a 1 ,... , an) + t[det(b 1 , a 2 ,... , an) + det(a 1 , b 2 ,... , an) t
- · · · + det(a 1 ,... , an− 1 , bn)] + t^2 (... ) − det(a 1 ,... , an) t = det(b 1 , a 2 ,... , an) + · · · + det(a 1 ,... , an− 1 , bn)
It is easy to show that 1 is a regular value of df (it suffices to show that df (A) is nonzero for any A ∈ SL(n, R)). For example, take b 1 = ca 1 where c ∈ R and bi = 0 for all i 6 = 1.
Theorem 7.6 (Sard’s theorem). Let f : U → V be a smooth map. Then almost every point y ∈ Rn is a regular value.
The notion of almost every point will be made precise later. But in the meantime:
Reality Check: In Sard’s theorem what happens when m < n?
DIFFERENTIAL GEOMETRY COURSE NOTES 19
Zen: The implicit function theorem tells us that under a constant rank condition we may assume that locally we can straighten our manifolds and maps and pretend we are doing linear algebra.
Examples of immersions:
(1) Circle mapped to figure 8 in R^2. (2) The map f : R → C, t 7 → eit, which wraps around the unit circle S^1 ⊂ C infinitely many times. (3) The map f : R → R^2 /Z^2 , t 7 → (at, bt), where b/a is irrational. The image of f is dense in R^2 /Z^2.
8.3. Embeddings and submanifolds. We upgrade immersions f : M → N as follows:
Definition 8.3. An embedding f : M → N is an immersion which is one-to-one and proper. The image of an embedding is called a submanifold of N.
The “pathological” examples above are immersions but not embeddings. Why? (1) and (2) are not one-to-one and (3) is not proper.
Proposition 8.4. Let M and N be manifolds of dimension m and n with topologies T and T ′. If f : M → N is an embedding, then f −^1 (T ′) = T.
Proof. It suffices to show that f −^1 (T ′) ⊃ T , since a continuous map f satisfies f −^1 (T ′) ⊂ T. Let x ∈ M and U be a small open set containing x. Then by the implicit function theorem f can be written locally as U → Rn, x′^7 → (x′, 0) (where we are using x′^ to avoid confusion with x). We claim that there is an open set V ⊂ Rn^ such that V ∩ f (M ) = f (U ): Arguing by contradiction, suppose there exist y ∈ f (U ) and a sequence {xi}∞ i=1 ⊂ M such that f (xi) → y but f (xi) 6 ∈ f (U ). The set {y} ∪ {f (xi)}∞ i=1 is compact, so {f −^1 (y)} ∪ {xi}∞ i=1 is compact by properness, where we are recalling that f is one-to-one. By compactness, there is a subsequence of {xi} which converges to f −^1 (y). This implies that xi ∈ U and f (xi) ∈ f (U ) for sufficiently large i, a contradiction.
20 KO HONDA
- TANGENT SPACES, DAY I
9.1. Concrete example. Consider S^2 = {x^2 + y^2 + z^2 = 1} ⊂ R^3. We recall the defini- tion/computation of the tangent plane T(a,b,c)S^2 from multivariable calculus. We use the fact that S^2 is the preimage of the regular value 1 of f , where
f : R^3 → R, f (x, y, z) = x^2 + y^2 + z^2.
The derivative of f at the point (a, b, c) is:
df (a, b, c)(x, y, z)T^ = (2a, 2 b, 2 c)(x, y, z)T^.
The tangent directions are directions (x, y, z) where df (a, b, c)(x, y, z)T^ = 0. If you want to think of the tangent plane as a vector space, then T(a,b,c)S^2 = {ax + by + cz = 0}. If you want to think of it as an affine space, then the tangent plane is the plane through (a, b, c) which is parallel to ax + by + cz = 0, i.e., T(a,b,c)S^2 = {ax + by + cz = a^2 + b^2 + c^2 = 1}.
Definition 9.1. Let M be a (codimension m) submanifold of Rn. Then we can define TpM as follows: Pick a small neighborhood Up ⊂ Rn^ of p and a submersion f : Up → Rm^ such that M ∩ Up is a level set of f. Then TpM is the set of vectors v ∈ Rn^ such that df (p)(v) = 0.
Some issues with this definition: (1) Need to verify that TpM does not depend on the choice of f : Up → Rm. (Not so serious.) (2) The definition seems to depend on how M is embedded in Rn. In other words, the definition is not intrinsic. We will give several definitions of TpM which are intrinsic, in increasing order of abstraction!!
9.2. First definition. Let M be a smooth n-dimensional manifold. If U ⊂ M is an open set, then let C∞(U ) be the set of smooth functions f : U → R.
Notation: Let f, g : U → R where 0 ∈ U ⊂ R. Then f = O(g) if there exists a constant C such that |f (t)| ≤ C|g(t)| for all t sufficiently close to 0. For example, t = sin t + O(t^3 ) near t = 0.
Definition 9.2 (First definition). The tangent space T (^) p(1) M (here (1) is to indicate that it’s the first definition) to M at p is the set of equivalence classes
T (^) p(1) (M ) = {smooth curves γ : (−εγ , εγ ) → M, γ(0) = p}/ ∼,
where γ 1 ∼ γ 2 if f ◦γ 1 (t) = f ◦γ 2 (t)+O(t^2 ) for all pairs (f, U ) where U is an open set containing p and f ∈ C∞(U ). Here εγ > 0 is a constant which depends on γ.
Let x 1 ,... , xn be coordinate functions for an open set U ⊂ M.
Theorem 9.3 (Taylor’s Theorem). Let f : U ⊂ Rn^ → R be a smooth function and 0 ∈ U. Then we can write f (x) = a +
i
aixi +
i,j
aij (x)xixj ,
on an open rectangle (−a 1 , b 1 ) × · · · × (−an, bn) ⊂ U which contains 0 , where a, ai are constants and aij (x) are smooth functions.
