Covering Spaces and the Fundamental Group of the Circle
A loop can wind around a hole even when every small neighbourhood looks Euclidean. The fundamental group records which loops can be deformed into one another while their base point stays fixed. Covering spaces help compute this group by replacing loops with paths upstairs.
For background, see (Hatcher 2002). We develop path homotopy and lifting, then compute the fundamental group of the circle by recording the endpoint of a lifted loop. The same winding-number argument gives applications to the fundamental theorem of algebra and the two-dimensional Borsuk–Ulam theorem.
Definition 1 (Homotopy). Let \(c_1 : [0,a_1] \to M\) and \(c_2 : [0,a_2] \to M\) be paths (continuous maps), with positive interval lengths, on a differentiable manifold \(M\). \(c_1\) is said to be a loop if it is a path with \(c_1(0) = c_1(a_1)\); \(c_1(0)\) is said to be the base point of the loop. \(c_1\) and \(c_2\) are said to be homotopic (as paths) if \(c_1(0)=c_2(0)\) and \(c_1(a_1)=c_2(a_2)\), and there exists a continuous map \(H : [0,1]^2 \to M\) such that
We write \(c_1 \simeq c_2\) to mean \(c_1\) and \(c_2\) are homotopic (as paths).
It is clear from the definition of homotopy that \(H\) is a continuous deformation of path \(c_1\) into path \(c_2\) that keeps endpoints fixed. Formally, the functions \(H_s (t) = H(s,t)\) generated by the path homotopy \(H\) are paths from \(c_1(0)\) to \(c_2(a_2)\). As a remark, distinctions are drawn between homotopy of continuous functions and homotopy of paths; here, we always mean homotopy of paths without explicit mention.
Lemma 2. \(\simeq\) is an equivalence relation on the space of paths between fixed endpoints.
Transitivity requires joining two homotopies. The pasting lemma ensures that the joined map is continuous when the pieces agree along their common boundary.
Lemma 3 (Pasting Lemma). Let \(f:X \to Y\) be a map between topological spaces \(X\) and \(Y\). Let \(X = A \cup B\), where \(A\) and \(B\) are both open or closed subsets of \(X\). If \(f|_A\) and \(f|_B\) are both continuous maps, then \(f\) is a continuous map.
Proof of the pasting lemma. If \(A,B\) are open, the inverse image of an open set is the union of two sets open in \(X\). If \(A,B\) are closed, use a closed set \(C\subset Y\): each \((f|_A)^{-1}(C)\) and \((f|_B)^{-1}(C)\) is closed in its closed domain and hence in \(X\), and their finite union is \(f^{-1}(C)\). In both cases \(f\) is continuous. The argument uses only that finite unions of closed sets are closed and that arbitrary unions of open sets are open. \(\square\)
Proof of Lemma 2. Normalize all paths to \(I=[0,1]\). Reflexivity uses \(H(s,t)=c(t)\); symmetry replaces \(s\) by \(1-s\). If \(F\) deforms \(f\) to \(g\) and \(G\) deforms \(g\) to \(h\), define
The two continuous pieces agree at \(s=1/2\), where both equal \(g(t)\), so the closed-set pasting lemma applies. The boundary values at \(t=0,1\) stay fixed. Thus \(f\simeq h\). \(\square\)
For a path \(c:[0,a]\to M\), \(a>0\), the normalized path on \(I\) is \(\widetilde c(t)=c(at)\). More generally, a continuous reparametrization \(\rho:I\to I\) fixing zero and one does not change a path class: \(c((1-s)t+s\rho(t))\) is an endpoint-fixing homotopy from \(c\) to \(c\circ\rho\).
Definition 4. The homotopy classes of loops with a fixed base point \(p \in M\) is called the fundamental group with base point \(p\), denoted \(\pi_1 (M,p)\).
When two points lie in the same path component, their fundamental groups are isomorphic, but the isomorphism requires a chosen connecting path. We use the convention that \(f\star g\) traverses \(f\) first and then \(g\). Thus, for paths \(f:x_0\to x_1\) and \(g:x_1\to x_2\), define
Clearly, \(f \star g\) is continuous by the pasting lemma, so \(f \star g\) is a path from \(x_0\) to \(x_2\). Applied to homotopy class, we write \([f] \star [g] = [f \star g]\).
Proposition 5. The operation \(\star\) between homotopy classes is well-defined.
Proof of Proposition 5 and the group laws. Given endpoint-fixing homotopies \(F\) from \(f\) to \(\widetilde f\) and \(G\) from \(g\) to \(\widetilde g\), use \(F(s,2t)\) for \(t\le1/2\) and \(G(s,2t-1)\) for \(t\ge1/2\). They agree at the common endpoint, proving \(f\star g\simeq\widetilde f\star\widetilde g\).
The paths \((f\star g)\star h\) and \(f\star(g\star h)\) traverse the same three paths, with breakpoints \((1/4,1/2)\) and \((1/2,3/4)\). Interpolate these two breakpoints linearly and traverse each path at constant speed on its resulting subinterval. All three interval lengths stay positive; the formulas agree at moving boundaries, giving an endpoint-fixing homotopy and hence associativity of classes. A constant path is an identity by the reparametrization homotopy above. For the reverse \(f^*(t)=f(1-t)\), put \(u(t)=2t\) on \([0,1/2]\) and \(u(t)=2-2t\) on \([1/2,1]\). Then \(f((1-s)u(t))\) contracts \(f\star f^*\) to the constant at \(f(0)\) while fixing endpoints. Applying this to \(f^*\) proves the other inverse law. \(\square\)
The operation \(\star\) makes homotopy classes of loops based at \(p_0\) into the group \(\pi_1(M,p_0)\). Let \(p:[0,1]\to M\) be a chosen path from \(p_0\) to \(p_1\). Its reverse is \(p^*(s)=p(1-s)\). Change of base point is the map \(\widehat p:\pi_1(M,p_0)\to\pi_1(M,p_1)\) given by
If a loop changes by a homotopy fixing its endpoints, concatenate that homotopy with \(p^*\) and \(p\). The resulting loops remain based at \(p_1\), so \(\widehat p\) is well-defined. Concatenation is associative on homotopy classes, and \([p\star p^*]\) and \([p^*\star p]\) are the appropriate constant-loop classes.
Proposition 6. \(\widehat p\) is an isomorphism.
Proof. First, \(\widehat p\) is a homomorphism between groups. Let \([f], [g] \in \pi_1(M,p_0)\). Then, by associativity of \(\star\),
Then, note that if \(\widetilde g \in \pi_1(M,p_1)\), we have that \([p] \star [\widetilde g] \star [p^*]\) is the homotopy class of a loop in \(M\) based at \(p_0\). Then it follows
This shows the surjectivity of \(\widehat p\). For the injectivity of \(\widehat p\), suppose \(\widehat p([f]) = \widehat p([g])\). It follows that
implies
Therefore, \(\widehat p\) is injective, and this concludes the proof. ◻
For another path \(q:p_0\to p_1\), let \(h=q^*\star p\), a loop at \(p_1\). Direct cancellation of reverse paths gives \(\widehat q([f])=[h]\star\widehat p([f])\star[h]^{-1}\). The two identifications therefore differ by an inner automorphism. This is why omitting the base point records the group only up to isomorphism; in an abelian fundamental group the inner automorphism is trivial.
We can then conclude the following.
Corollary 7. For \(p_0, p_1 \in M\), if there exists a path connecting \(p_0\) and \(p_1\), then \(\pi_1(M,p_0)\) and \(\pi_1(M,p_1)\) are isomorphic.
Definition 8. A manifold \(M\) is simply connected if it is path-connected and its fundamental group \(\pi_1(M,p)\) is trivial for one, equivalently every, base point \(p\).
Remark 9. A continuous map \(f:M\to N\) induces a group homomorphism
Composing a path homotopy with \(f\) shows this is well-defined. Since \(f\circ(c\star d)=(f\circ c)\star(f\circ d)\), it respects multiplication. It need not be injective or surjective: a constant map sends every loop class to the identity. A homeomorphism does induce an isomorphism, because its inverse induces the inverse homomorphism.
For a connected manifold \(M\), the covering-space existence theorem states that each subgroup \(H\le\pi_1(M,p)\) is realized by a connected covering \(\pi:X\to M\) and a chosen point \(x\in\pi^{-1}(p)\) with \(\pi_\#\pi_1(X,x)=H\). The manifold is locally path-connected and semilocally simply connected, as required by Hatcher, Proposition 1.36. Other points of the fibre can give conjugate subgroups, so equality with \(H\) is not asserted at every lift. For \(H=\{1\}\), the connected covering is simply connected and is called the universal covering.
We now turn to covering spaces. Once the starting point above a path is chosen, local inverse maps determine its lift; the same principle applies to homotopies.
Definition 10. Let \(\pi : X \to M\) be a covering of \(M\), \(c : [0,1] \to M\) be a path, and \(x_0 \in \pi^{-1} (c(0))\). Then, there exists a unique path \(\widetilde c: [0,1] \to X\) such that \(\widetilde c(0) = x_0\) and \(c(t) = \pi(\widetilde c(t))\). \(\widetilde c\) is called the lift of \(c\) through \(x_0\).
Remark 11. (Lifting criterion.) Let \(M'\) be a connected manifold, \(f:(M',y_0)\to(M,p_0)\) continuous, and \(\pi:(X,x_0)\to(M,p_0)\) a covering. A lift \(\widetilde f\) with \(\widetilde f(y_0)=x_0\) exists exactly when
It is then unique. The group on the right belongs to the covering space \(X\), not the base \(M\).
Proof, using the lifting lemma proved below. Necessity follows by applying fundamental groups to \(f=\pi\widetilde f\). A loop at \(p_0\) lifts to a loop at \(x_0\) exactly when its class lies in \(\pi_\#\pi_1(X,x_0)\): one implication follows by projection; the other follows by lifting an endpoint-fixing homotopy from a projected loop. Under the stated inclusion, choose a path \(a\) from \(y_0\) to \(y\) and define \(\widetilde f(y)\) as the endpoint of the lift of \(f\circ a\) from \(x_0\). For two such paths \(a,b\), the loop \(a\star b^*\) has image lifting to a loop, so the two endpoints agree. This defines \(\widetilde f\) independently of choices.
Near \(y\), choose a path-connected neighbourhood \(W\) mapped by \(f\) into an evenly covered set \(U\). Paths from \(y\) to points of \(W\) show that \(\widetilde f|_W=(\pi|_{\widetilde U})^{-1}\circ f\) for the sheet containing \(\widetilde f(y)\), proving continuity. Any lift must have these same endpoints along paths from \(y_0\), proving uniqueness. Connected manifolds are path-connected and locally path-connected, supplying exactly the properties used here. For disconnected domains one must specify data and check the criterion on each component. \(\square\)
The same lifting argument proves that \(\pi_\#\) is injective: a loop upstairs whose projection is null-homotopic lifts that null-homotopy, so is itself null-homotopic.
Definition 12. Let \(\pi : \widetilde M \to M\) be the universal covering of \(M\). A deck transformation is a homeomorphism \(\varphi : \widetilde M \to \widetilde M\) such that \(\pi = \pi \circ \varphi\).
For a connected universal cover and a chosen \(x_0\) above \(p_0\), lifting gives a bijection between \(\pi_1(M,p_0)\) and the fibre \(\pi^{-1}(p_0)\). Surjectivity follows by projecting a path upstairs from \(x_0\) to a chosen fibre point. Two projected paths give the same class exactly when their lifts have the same endpoint: in the converse direction, concatenate one lift with the reverse of the other and use simple connectedness upstairs.
The group of deck transformations, with composition, acts freely and transitively on this fibre. Indeed, the lifting criterion applied to the map \(\pi:\widetilde M\to M\) gives a unique lift \(T:\widetilde M\to\widetilde M\) taking \(x_0\) to any prescribed point \(x'\) of the fibre. A lift in the reverse direction exists as well; uniqueness makes the two composites identities, so \(T\) is a homeomorphism. A deck map fixing \(x_0\) is the identity by uniqueness. Under the path model, the deck map associated with a loop \(a\) sends \([\gamma]\) to \([a\star\gamma]\), so \(T_a\circ T_b=T_{a\star b}\) with our convention for path products. This identifies the deck group with \(\pi_1(M,p_0)\). The fibre is a set on which this group acts freely and transitively; it has no distinguished identity until a lift \(x_0\) is chosen. A single deck transformation need not act transitively by itself.
Example 13 (Fundamental groups). Fundamental groups are notoriously difficult to determine. In simple spaces, we can explicitly determine them. For example, \(\pi_1(\mathbb R^n) = \{1\}\), \(\pi_1(\mathbb S^n) = \{1\}\) for \(n \geq 2\), \(\pi_1(\operatorname{SO}(n)) = \mathbb Z_2\) for \(n \geq 3\). Perhaps the most interesting, \(\pi_1(\mathbb S^1) = \mathbb Z\), and the generator is \(c(t) = (\cos 2\pi t, \sin 2\pi t)\); the universal covering of \(\mathbb S^1\) is \(\mathbb R\).
For \(\mathbb R^n\), the straight-line homotopy \((1-s)c(t)+sc(0)\) contracts every based loop. For \(\mathbb S^n\), \(n\ge2\), subdivide a loop into pieces lying in small geodesically convex balls and homotope each piece, fixing its endpoints, to the short geodesic between them. The resulting finite union of arcs misses a point of the sphere. Stereographic projection from that point reduces its contraction to the Euclidean case, keeping the base point fixed.
For the additional example \(\operatorname{SO}(n)\), \(n\ge3\), a standard external input is its CW structure: the two-skeleton is \(\mathbb{RP}^2\), and remaining cells have dimension at least three (Hatcher, Proposition 3D.1, pp. 295–296). Thus its presentation has one generator and the relation that its square is the identity; higher-dimensional cells do not alter \(\pi_1\). This gives \(\mathbb Z/2\mathbb Z\).
1. Fundamental Group of \(\mathbb S^1\)
Algorithmic statements require a specified finite input. The Adian–Rabin theorem implies that no algorithm decides whether the group specified by an arbitrary finite presentation is trivial; see Adian’s papers in Nyberg-Brodda’s translation, theorem and example (iii), p. 3. For each fixed dimension \(n\ge4\), effective constructions realize such presentations as fundamental groups of closed triangulated \(n\)-manifolds, so deciding simple connectedness in that class is also impossible. The construction uses a presentation two-complex and the boundary of a regular neighbourhood in \(\mathbb R^{n+1}\); see Soare, §3.3, pp. 7–8 of the December 2003 manuscript. This does not assert undecidability in every dimension or for every restricted family of manifolds. We now compute the circle’s group explicitly.
Theorem 14. \(\pi_1(\mathbb S^1)\) is an infinite cyclic group generated by homotopy class of the loop \(c(t) = (\cos 2\pi t, \sin 2\pi t)\) with base point \((1,0)\).
With the ongoing notations, note that \([c]^n = [c_n] = [(\cos 2\pi nt, \sin 2\pi nt)]\) for all \(n \in \mathbb Z\). Therefore, the statement of Theorem 14 is equivalent to showing that every loop in \(\mathbb S^1\) based at \((1,0)\) is homotopic to \(c_n\) for some unique \(n \in \mathbb Z\). The strategy of proving this equivalent statement is the following. Define \(\widetilde c_n(t) = nt\), \(c_n = p \circ \widetilde c_n\), where \(p\) is the projection of the embedding \(t \mapsto (\cos 2\pi t, \sin 2\pi t, t)\) onto the first two coordinates. Clearly, \(p\) is a covering map for \(\mathbb S^1\). Then, we need to show the following properties of a covering map \(p : \widetilde M \to M\) (for any manifold \(M\)): for all paths \(f : I \to M\) starting at \(x_0 \in M\), and for all \(\widetilde x_0 \in p^{-1}(x_0)\), there exists a unique lift \(\widetilde f : I \to \widetilde M\) such that \(\widetilde f\) starts at \(\widetilde x_0\). Additionally, for all homotopy fixed at \(s\), denoted \(H_s : I \to X : (s,t) \mapsto c_s(t)\), where each curve in a homotopy class starts at \(x_0 \in X\), for all \(\widetilde x_0 \in p^{-1}(x_0)\), there exists a unique lifted homotopy \(\widetilde H_s\) of paths starting at \(\widetilde x_0\). We deduce these facts from a more general claim.
Lemma 15. Let \(p:\widetilde M\to M\) be a covering, \(Y\) any topological space, and \(F:Y\times I\to M\) continuous. Given a continuous lift \(a:Y\to\widetilde M\) of \(F(\cdot,0)\), there is a unique continuous \(\widetilde F:Y\times I\to\widetilde M\) such that \(p\widetilde F=F\) and \(\widetilde F(y,0)=a(y)\).
Proof. Fix \(y_0\). By continuity and compactness of \(I\), there are a neighbourhood \(N\) of \(y_0\) and a subdivision \(0=t_0<\cdots<t_m=1\) such that every \(F(N\times[t_i,t_{i+1}])\) lies in an evenly covered open set \(U_i\). To obtain this subdivision, cover \(\{y_0\}\times I\) by finitely many product neighbourhoods with this property, use a Lebesgue number for their time intervals, and intersect their neighbourhoods in \(Y\).
At \(t_0\), choose the sheet \(\widetilde U_0\) containing \(a(y_0)\) and shrink \(N\) so that \(a(N)\subset\widetilde U_0\). Define the lift on the first strip by \((p|_{\widetilde U_0})^{-1}\circ F\). At step \(i\), the already constructed boundary value \(y\mapsto\widetilde F(y,t_i)\) is continuous. Choose the sheet over \(U_i\) containing its value at \(y_0\) and shrink \(N\) so that all these boundary values lie in that sheet. Its inverse branch defines the next strip and agrees on the shared boundary. Finitely many shrinkings and the pasting lemma produce a continuous lift on \(N\times I\).
For a single path, two lifts with the same start stay in the same sheet on each such subinterval: a connected path image contained in the disjoint union of sheets lies entirely in one sheet. Injectivity of \(p\) on that sheet gives equality, successively on all subintervals. Thus lifts are unique pointwise on each \(\{y\}\times I\). Consequently the constructions for different neighbourhoods \(N\) agree on every overlap and glue to a continuous global lift. The same path uniqueness proves global uniqueness. \(\square\)
Corollary 16. Every path has a unique lift from a prescribed starting point. Every endpoint-fixing homotopy \(H(s,t)\) of paths has a unique lift whose paths start at a prescribed lift \(\widetilde x_0\) of the common initial point; the endpoints of the lifted paths are independent of \(s\).
Proof. The first assertion is Lemma 15 with \(Y\) a point. For the second, take \(Y=I\) with coordinate \(s\), use \(t\) as the lifting time, and prescribe \(\widetilde H(s,0)=\widetilde x_0\). The map \(s\mapsto\widetilde H(s,1)\) is continuous and lies in the discrete fibre above the common terminal point, so it is constant. The constant edges are \(I\times\{0\}\) and \(I\times\{1\}\), not the two edges parametrizing the original paths. \(\square\)
Proof of Theorem 14. Write \(p(u)=e^{2\pi iu}\). Short open arcs in the circle have inverse images consisting of disjoint translates of an interval of length less than one, on each of which \(p\) is a homeomorphism. Thus \(p:\mathbb R\to\mathbb S^1\) is a covering, and it is universal because \(\mathbb R\) is simply connected.
For a loop \(f\) based at \(1\), let \(\widetilde f\) be its lift from zero. Its endpoint \(n=\widetilde f(1)\) is an integer. Define
This is a homotopy from \(f\) to \(c_n(t)=e^{2\pi int}\), with \(H(s,0)=H(s,1)=1\). If \(f\) and \(g\) are based-homotopic, Corollary 16 makes their lifted endpoints equal. Conversely, equal endpoints give the displayed linear homotopy. Hence the endpoint gives a bijection \(w:\pi_1(\mathbb S^1,1)\to\mathbb Z\).
To check the group law explicitly, if \(w(f)=m\) and \(w(g)=n\), a lift of \(f\star g\) is \(\widetilde f(2t)\) for \(t\le1/2\) and \(m+\widetilde g(2t-1)\) for \(t\ge1/2\). Its endpoint is \(m+n\), so \(w\) is an isomorphism. The once-around loop has endpoint one and generates the group. \(\square\)
For loops at any other point of the circle, the same integer is the difference between terminal and initial lift values; multiplication by a fixed unit complex number reduces this to the base point \(1\).
Theorem 14 has the following applications.
Theorem 17 (Fundamental theorem of algebra). Every nonconstant polynomial with coefficients in \(\mathbb C\) has a root in \(\mathbb C\).
Proof. Divide by the leading coefficient and write \(p(z)=z^n+a_1z^{n-1}+\cdots+a_n\) with \(n\ge1\). Suppose \(p\) never vanishes. For \(r\ge0\), put
The denominator is nonzero, and \(f_r(0)=f_r(1)=1\). For fixed \(R\), \((s,t)\mapsto f_{sR}(t)\) is a continuous based homotopy from the constant loop \(f_0\) to \(f_R\), so \(w(f_R)=0\).
Choose \(R>\max\{1,\sum_j|a_j|\}\). On \(|z|=R\),
Thus \(p_s(z)=z^n+s\sum_j a_jz^{n-j}\) never vanishes on this circle for \(0\le s\le1\), including at \(z=R\). Replacing \(p\) by \(p_s\) in the formula for \(f_R\) therefore gives a based homotopy from \(c_n\) at \(s=0\) to \(f_R\) at \(s=1\). This is a homotopy of loops, not a deformation retraction. It gives \(n=w(c_n)=w(f_R)=0\), contradicting \(n\ge1\). \(\square\)
Theorem 18 (Borsuk-Ulam; dimension \(2\)). For every continuous map \(f : \mathbb S^2 \to \mathbb R^2\), there exists a pair of antipodal points \(x\) and \(-x\) in \(\mathbb S^2\) with \(f(x)=f(-x)\).
Remark 19. The general Borsuk–Ulam theorem (an external result; see Hatcher, Corollary 2B.7, p. 176) is true for any dimension \(n \geq 1\). In dimension \(1\), the statement is straightforward to verify, since the difference \(f(x)- f(-x)\) changes sign as \(x\) goes halfway around the circle, hence this difference must be zero for some \(x\) by the intermediate value theorem. For \(n \geq 2\), the theorem is certainly less obvious. The theorem also says in particular that there is no one-to-one continuous map from \(\mathbb S^2\) to \(\mathbb R^2\), so \(\mathbb S^2\) is not homeomorphic to a subspace of \(\mathbb R^2\). While heuristically true, this is hard to verify directly.
Proof of Theorem 18. If \(f(x)\ne f(-x)\) everywhere, then
is a continuous map \(\mathbb S^2\to\mathbb S^1\) satisfying \(g(-x)=-g(x)\). Restrict to the equator \(e(t)=(\cos2\pi t,\sin2\pi t,0)\) and put \(h=g\circ e\). Then \(h(t+1/2)=-h(t)\) for \(0\le t\le1/2\). Lift \(h\) to \(\widetilde h:I\to\mathbb R\). Since \(p(u+1/2)=-p(u)\),
This difference is continuous on a connected interval and takes values in a discrete set, so it is a constant \(r/2\) with \(r\) odd. Applying it at \(t=0\) and \(t=1/2\) gives \(\widetilde h(1)-\widetilde h(0)=r\). Thus \(h\) has nonzero winding number.
On the other hand, the equator bounds the upper hemisphere, a disk. Its boundary loop contracts in that disk while fixing its base point: identify the hemisphere with the closed unit disk and use the linear homotopy from the boundary loop to its initial boundary point. Composing this homotopy with \(g\) makes \(h\) based null-homotopic, so its winding number is zero. This contradiction proves the claim. \(\square\)
Proposition 20. \(\pi_1(X \times Y)\) is isomorphic to \(\pi_1(X) \times \pi_1(Y)\) if \(X\) and \(Y\) are path-connected.
Proof. This is the consequence of a fact about product topology: a map \(f: Z \to X \times Y\), defined by \(f(z) = (g(z), h(z))\), is continuous if and only if the maps \(g: Z \to X\) and \(h: Z \to Y\) are both continuous. Therefore, a loop \(f\) in \(X \times Y\) based at \((x_0, y_0)\) is equivalent to a pair of loops \(g\) in \(X\) and \(h\) in \(Y\) based at \(x_0\) and \(y_0\), respectively.
Similarly, a homotopy \(f_s\) of a loop in \(X \times Y\) is equivalent to a pair of homotopies \(g_t\) and \(h_t\) of the corresponding loops \(g\) and \(h\) in \(X\) and \(Y\), respectively. Therefore, we have a bijection that maps \([f]\) to \(([g],[h])\). The inverse sends a pair of loop classes to the class of their pointwise pair. Endpoint-fixing homotopies pair in the same way, so both maps are well-defined. Concatenation is coordinatewise, which proves that the bijection is a group homomorphism. ◻
Reading: Carmo (1992), Riemannian geometry.