Basic Riemannian Geometry

F.E. Burstall Department of Mathematical Sciences University of Bath

Introduction

My mission was to describe the basics of Riemannian geometry in just three hours of lectures, starting from scratch. The lectures were to provide background for the analytic matters covered elsewhere during the conference and, in particular, to underpin the more detailed (and much more professional) lectures of Isaac Chavel. My strategy was to get to the point where I could state and prove a Real Live Theorem: the Bishop Volume Comparison Theorem and Gromov’s improvement thereof and, by appalling abuse of OHP technology, I managed this task in the time alloted. In writing up my notes for this volume, I have tried to retain the breathless quality of the original lectures while correcting the mistakes and excising the out-right lies.

I have given very few references to the literature in these notes so a few remarks on sources is appropriate here. The first part of the notes deals with analysis on differentiable manifolds. The two canonical texts here are Spivak [5] and Warner [6] and I have leaned on Warner’s book in particular. For Riemannian geometry, I have stolen shamelessly from the excellent books of Chavel [1] and Gallot–Hulin–Lafontaine [3]. In particular, the proof given here of Bishop’s theorem is one of those provided in [3].

1 What is a manifold?

What ingredients do we need to do Differential Calculus? Consider first the notion of a continuous function: during the long process of abstraction and generalisation that leads from Real Analysis through Metric Spaces to Topology, we learn that continuity of a function requires no more structure on the domain and co-domain than the idea of an open set.

By contrast, the notion of differentiability requires much more: to talk about the difference quotients whose limits are partial derivatives, we seem to require that the (co-)domain have a linear (or, at least, affine) structure.

However, a moment’s thought reveals that differentiability is a completely local matter so that all that is really required is that the domain and codomain be locally linear, that is, each point has a neighbourhood which is homeomorphic to an open subset of some linear space. These ideas lead us to the notion of a manifold: a topological space which is locally Euclidean and on which there is a well-defined differential calculus.

We begin by setting out the basic theory of these spaces and how to do Analysis on them.

1.1 Manifolds

Let M be a Hausdorff, second countable^{1}, connected topological space.

M is a C^{r} manifold of dimension n if there is an open cover {U_{α}}_{α∈I} of M and homeomorphisms x_{α} : U_{α} → x_{α}(U_{α}) onto open subsets of ℝ^{n} such that, whenever U_{α} ∩ U_{β} ≠ ∅,

x_{α} ∘ x_{β}^{−1} : x_{β}(U_{α} ∩ U_{β}) → x_{α}(U_{α} ∩ U_{β})

is a C^{r} diffeomorphism.

Each pair (U_{α}, x_{α}) called a chart.

Write x_{α} = (x^{1}, …, x^{n}). The x^{i} : U_{α} → ℝ are coordinates.

1.1.1 Examples

1. Any open subset U ⊂ ℝ^{n} is a C^{∞} manifold with a single chart (U, 1_{U}).

2. Contemplate the unit sphere S^{n} = {v ∈ ℝ^{n+1} : ∥v∥ = 1} in ℝ^{n+1}. Orthogonal projection provides a homeomorphism of any open hemisphere onto the open unit ball in some hyperplane ℝ^{n} ⊂ ℝ^{n+1}. The sphere is covered by the (2n + 2) hemispheres lying on either side of the coordinate hyperplanes and in this way becomes a C^{∞} manifold (exercise!).

3. A good supply of manifolds is provided by the following version of the Implicit Function Theorem [6]:

Theorem. Let f : Ω ⊂ ℝ^{n} → ℝ be a C^{r} function (r ≥ 1) and c ∈ ℝ a regular value, that is, ∇f(x) ≠ 0, for all x ∈ f^{−1}{c}.

Then f^{−1}{c} is a C^{r} manifold.

Exercise. Apply this to f(x) = ∥x∥^{2} to get a less tedious proof that S^{n} is a manifold.

1This means that there is a countable base for the topology of M.

4. An open subset of a manifold is a manifold in its own right with charts (U_{α} ∩ U, x_{α}|_{U_{α}∩U}).

1.1.2 Functions and maps

A continuous function f : M → ℝ is C^{r} if each f ∘ x_{α}^{−1} : x_{α}(U_{α}) → ℝ is a C^{r} function of the open set x_{α}(U_{α}) ⊂ ℝ^{n}.

We denote the vector space of all such functions by C^{r}(M).

Example. Any coordinate function x^{i} : U_{α} → ℝ is C^{r} on U_{α}.

Exercise. The restriction of any C^{r} function on ℝ^{n+1} to the sphere S^{n} is C^{r} on S^{n}.

In the same way, a continuous map ϕ : M → N of C^{r} manifolds is C^{r} if, for all charts (U, x), (V, y) of M and N respectively, y ∘ ϕ ∘ x^{−1} is C^{r} on its domain of definition.

A slicker formulation^{2} is that h ∘ ϕ ∈ C^{r}(M), for all h ∈ C^{r}(M).

At this point, having made all the definitions, we shall stop pretending to be anything other than Differential Geometers and henceforth take r = ∞.

1.2 Tangent vectors and derivatives

We now know what functions on a manifold are and it is our task to differentiate them. This requires some less than intuitive definitions so let us step back and remind ourselves of what differentiation involves.

Let f : Ω ⊂ ℝ^{n} → ℝ and contemplate the derivative of f at some x ∈ Ω. This is a linear map df_{x} : ℝ^{n} → ℝ. However, it is better for us to take a dual point of view and think of v ∈ ℝ^{n} is a linear map v : C^{∞}(M) → ℝ by

vf =^{def} df_{x}(v).

The Leibniz rule gives us

v(fg) = f(x)v(g) + v(f)g(x).  (1.1)

Fact. Any linear v : C^{∞}(Ω) → ℝ satisfying (1.1) arises this way.

Now let M be a manifold. The preceding analysis may give some motivation to the following Definition. A tangent vector at m ∈ M is a linear map ξ : C^{∞}(M) → ℝ such that

2It requires a little machinery, in the shape of bump functions, to see that this is an equivalent formulation.

ξ(fg) = f(m)ξ(g) + ξ(f)g(m)

for all f, g ∈ C^{∞}(M).

Denote by M_{m} the vector space of all tangent vectors at m.

Here are some examples

1. For γ : I → M a (smooth) path with γ(t) = m, define γ′(t) ∈ M_{m} by

γ′(t)f = (f ∘ γ)′(t).

Fact. All ξ ∈ M_{m} are of the form γ′(t) for some path γ.

2. Let (U, x) be a chart with coordinates x^{1}, …, x^{n} and x(m) = p ∈ ℝ^{n}.

Define ∂_{i|m} ∈ M_{m} by

∂_{i|m}f = \frac{∂(f ∘ x^{−1})}{∂x^{i}} |_{p}

Fact. ∂_{1|m}, …, ∂_{n|m} is a basis for M_{m}.

3. For p ∈ U ⊂ ℝ^{n} open, we know that U_{p} is canonically isomorphic to ℝ^{n} via

vf = df_{p}(v)

for v ∈ ℝ^{n}.

4. Let M = f^{−1}{c} be a regular level set of f : Ω ⊂ ℝ^{n} → ℝ. One can show that M_{m} is a linear subspace of Ω_{m} ≅ ℝ^{n}. Indeed, under this identification,

M_{m} = {v ∈ ℝ^{n}: v ⊥ ∇f_{m}}.

Now that we have got our hands on tangent vectors, the definition of the derivative of a function as a linear map on tangent vectors is almost tautological:

Definition. For f ∈ C^{∞}(M), the derivative df_{m} : M_{m} → ℝ of f at m ∈ M is defined by

df_{m}(ξ) = ξf.

We note:

1. Each df_{m} is a linear map and the Leibniz Rule holds:

d(fg)_{m} = g(m)df_{m} + f(m)dg_{m}.

2. By construction, this definition coincides with the usual one when M is an open subset of ℝ^{n}.

Exercise. If f is a constant map on a manifold M, show that each df_{m} = 0.

The same circle of ideas enable us to differentiate maps between manifolds:

Definition. For ϕ : M → N a smooth map of manifolds, the tangent map dϕ_{m} : M_{m} → N_{ϕ(m)} at m ∈ M is the linear map defined by

dϕ_{m}(ξ)f = ξ(f ∘ ϕ),

for ξ ∈ M_{m} and f ∈ C^{∞}(N).

Exercise. Prove the chain rule: for ϕ : M → N and ψ : N → Z and m ∈ M,

d(ψ ∘ ϕ)_{m} = dψ_{ϕ(m)} ∘ dϕ_{m}.

Exercise. View ℝ as a manifold (with a single chart!) and let f : M → ℝ. We now have two competing definitions of df_{m}. Show that they coincide.

The tangent bundle of M is the disjoint union of the tangent spaces:

T M = ∐_{m∈M} M_{m}.

1.3 Vector fields

Definition. A vector field is a linear map X : C^{∞}(M) → C^{∞}(M) such that

X(fg) = f(Xg) + g(Xf).

Let Γ(T M) denote the vector space of all vector fields on M.

We can view a vector field as a map X : M → T M with X(m) ∈ M_{m}: indeed, we have

X_{|m} ∈ M_{p}

where

X_{|m}f = (Xf)(m).

In fact, vector fields can be shown to be exactly those maps X : M → T M with X(m) ∈ M_{m} which satisfy the additional smoothness constraint that for each f ∈ C^{∞}(M), the function m ↦ X(m)f is also C^{∞}.

The Lie bracket of X, Y ∈ Γ(T M) is [X, Y] : C^{∞}(M) → C^{∞}(M) given by

[X, Y]f = X(Y f) − Y (Xf).

The point of this definition is contained in the following

Exercise. Show that [X, Y] ∈ Γ(T M) also.

The Lie bracket is interesting for several reasons. Firstly it equips Γ(T M) with the structure of a Lie algebra; secondly, it, and operators derived from it, are the only differential operators that can be defined on an arbitrary manifold without imposing additional structures such as special coordinates, a Riemannian metric, a complex structure or a symplectic form.

There is an extension of the notion of vector field that we shall need later on:

Definition. Let ϕ : M → N be a map. A vector field along ϕ is a map X : M → T N with

X(m) ∈ N_{ϕ(m)},

for all m ∈ M, which additionally satisfies a smoothness assumption that we shall gloss over.

Denote by Γ(ϕ^{−1}T N) the vector space of all vector fields along ϕ.

Here are some examples:

1. If c : I → N is a smooth path then c′ ∈ Γ(ϕ^{−1}T N).

2. More generally, for ϕ : M → N and X ∈ Γ(T M), dϕ(X) ∈ Γ(ϕ^{−1}T N). Here, of course,

dϕ(X)(m) = dϕ_{m}(X_{|m}).

3. For Y ∈ Γ(T N), Y ∘ ϕ ∈ Γ(ϕ^{−1}T N).

1.4 Connections

We would like to differentiate vector fields but as they take values in different vector spaces at different points, it is not so clear how to make difference quotients and so derivatives. What is needed is some extra structure: a connection which should be thought of as a “directional derivative” for vector fields.

Definition. A connection on T M is a bilinear map

T M × Γ(T M) → T M
(ξ, X) ↦ ∇_{ξ}X

such that, for ξ ∈ M_{m}, X, Y ∈ Γ(T M) and f ∈ C^{∞}(M),

1. ∇_{ξ}X ∈ M_{m};

2. ∇_{ξ}(fX) = (ξf)X_{|m} + f(m)∇_{ξ}X;

3. ∇_{X}Y ∈ Γ(T M).

A connection on T M comes with some additional baggage in the shape of two multilinear maps:

T_{m} : M_{m} × M_{m} → M_{m}
R_{m} : M_{m} × M_{m} × M_{m} → M_{m}

given by

T_{m}(ξ, η) = ∇_{ξ}Y − ∇_{η}X − [X, Y]_{|m}
R_{m}(ξ, η)ζ = ∇_{η}∇_{X}Z − ∇_{ξ}∇_{Y} Z − ∇_{[Y,X]_{|m}}

where X, Y, Z ∈ Γ(T M) with X_{|m} = ξ, Y_{|m} = η and Z_{|m} = ζ.

T_{m} and R_{m} are, respectively, the torsion and curvature at m of ∇.

Fact. R and T are well-defined—they do not depend of the choice of vector fields X, Y and Z extending ξ, η and ζ.

We have some trivial identities:

T (ξ, η) = −T (η, ξ)
R(ξ, η)ζ = −R(η, ξ)ζ.

and, if each T_{m} = 0, we have the less trivial First Bianchi Identity:

R(ξ, η)ζ + R(ζ, ξ)η + R(η, ζ)ξ = 0.

A connection ∇ on T N induces a similar operator on vector fields along a map ϕ : M → N. To be precise, there is a unique bilinear map

T M × Γ(ϕ^{−1}T N) → T N
(ξ, X) ↦ ϕ^{−1}∇_{ξ}X

such that, for ξ ∈ M_{m}, X ∈ Γ(T M), Y ∈ Γ(ϕ^{−1}T N) and f ∈ C^{∞}(M),

1. ϕ^{−1}∇_{ξ}Y ∈ N_{ϕ(m)};

2. ϕ^{−1}∇_{ξ}(fY) = (ξf)Y_{|ϕ(m)} + f(m)ϕ^{−1}∇_{ξ}Y;

3. ϕ^{−1}∇_{X}Y ∈ Γ(ϕ^{−1}T N) (this is a smoothness assertion);

4. If Z ∈ Γ(T N) then Z ∘ ϕ ∈ Γ(ϕ^{−1}T N) and

ϕ^{−1}∇_{ξ}(Z ∘ ϕ) = ∇_{dϕ_{m}(ξ)}Z.

ϕ^{−1}∇ is the pull-back of ∇ by ϕ. The first three properties just say that ϕ^{−1}∇ behaves like ∇, it is the last that essentially defines it in a unique way.

2 Analysis on Riemannian manifolds

2.1 Riemannian manifolds

A rich and useful geometry arises if we equip each M_{m} with an inner product:

Definition. A Riemannian metric g on M is an inner product g_{m} on each M_{m} such that, for all vector fields X and Y, the function

m ↦ g_{m}(X_{|m}, Y_{|m})

is smooth.

A Riemannian manifold is a pair (M, g) with M a manifold and g a metric on M.

Here are some (canonical) examples:

1. Let ( , ) denote the inner product on ℝ^{n}.

An open U ⊂ ℝ^{n} gets a Riemannian metric via U_{m} ≅ ℝ^{n}:

g_{m}(v, w) = (v, w).

2. Let S^{n} ⊂ ℝ^{n+1} be the unit sphere. Then S_{m}^{n} ≅ m^{⊥} ⊂ ℝ^{n+1} and so gets a metric from the inner product on ℝ^{n+1}.

3. Let D^{n} ⊂ ℝ^{n} be the open unit disc but define a metric by

g_{z}(v, w) = \frac{4(v, w)}{(1 − |z|^{2})^{2}}

(D^{n}, g) is hyperbolic space.

Much of the power of Riemannian geometry comes from the fact that there is a canonical choice of connection. Consider the following two desirable properties for a connection ∇ on (M, g):

1. ∇ is metric: Xg(Y, Z) = g(∇_{X}Y, Z) + g(Y, ∇_{X}Z).

2. ∇ is torsion-free: ∇_{X}Y − ∇_{Y} X = [X, Y]

Theorem. There is a unique torsion-free metric connection on any Riemannian manifold.

Proof. Assume that g is metric and torsion-free. Then

g(∇_{X}Y, Z) = Xg(Y, Z) − g(Y, ∇_{X}Z)
= Xg(Y, Z) − g(Y, [X, Z]) − g(Y, ∇_{Z}X) …

and eventually we get

2g(∇_{X}Y, Z) = Xg(Y, Z) + Y g(Z, Y) − Zg(X, Y) − g(X, [Y, Z]) + g(Y, [Z, X]) + g(Z, [X, Y]). (2.1)

This formula shows uniqueness and, moreover, defines the desired connection.

This connection is the Levi–Civita connection of (M, g).

For detailed computations, it is sometimes necessary to express the metric and Levi–Civita connection in terms of local coordinates. So let (U, x) be a chart and ∂_{1}, …, ∂_{n} be the corresponding vector fields on U. We now define g_{ij} ∈ C^{∞}(U) by

g_{ij} = g(∂_{i}, ∂_{j})

and Christoffel symbols Γ_{ij}^{k} ∈ C^{∞}(U) by

∇_{∂_{i}}∂_{j} = ∑_{k} Γ_{ij}^{k}∂_{k}.

(Recall that ∂_{1|m}, …, ∂_{n|m} form a basis for M_{m}.)

Now let (g^{ij}) be the matrix inverse to (g_{ij}). Then the formula (2.1) for ∇ reads:

Γ_{ij}^{k} = \frac{1}{2} ∑_{l} g^{kl}(∂_{i}g_{jl} + ∂_{j}g_{li} − ∂_{l}g_{ij})  (2.2)

since the bracket terms [∂_{i}, ∂_{j}] vanish (exercise!).

2.2 Differential operators

The metric and Levi–Civita connection of a Riemannian manifold are precisely the ingredients one needs to generalise the familiar operators of vector calculus:

The gradient of f ∈ C^{∞}(M) is the vector field grad f such that, for Y ∈ Γ(T M),

g(grad f, Y) = Y f.

Similarly, the divergence of X ∈ Γ(T M) is the function div f ∈ C^{∞}(M) defined by:

(div f)(m) = trace(ξ → ∇_{ξ}X)

Finally, we put these together to introduce the hero of this volume: the Laplacian of f ∈ C^{∞}(M) is the function

Δf = div grad f.

In a chart (U, x), set g = det(g_{ij}). Then

grad f = ∑_{i,j} g^{ij}(∂_{i}f)∂_{j}

and, for X = ∑_{i} X_{i}∂_{i},

div X = ∑_{i}(∂_{i}X_{i} + ∑_{j} Γ_{ij}^{i}X_{j})

= \frac{1}{\sqrt{g}} ∑_{j} ∂_{j}(\sqrt{g}X_{j}).

Here we have used ∑_{i} Γ_{ij}^{i} = (∂_{j}\sqrt{g})/\sqrt{g} which the Reader is invited to deduce from (2.2) together with the well-known formula for a matrix-valued function A:

d ln det A = trace A^{−1}dA.

In particular, we conclude that

Δf = \frac{1}{\sqrt{g}} ∑_{i,j} ∂_{i}(\sqrt{g}g^{ij}∂_{j}f) = ∑_{i,j} g^{ij}(∂_{i}∂_{j}f − Γ_{ij}^{k}∂_{k}f).

2.3 Integration on Riemannian manifolds

2.3.1 Riemannian measure

(M, g) has a canonical measure dV on its Borel sets which we define in steps:

First let (U, x) be a chart and f : U → ℝ a measureable function. We set

∫_{U} f dV = ∫_{x(U)}(f ∘ x^{−1})\sqrt{g ∘ x^{−1}} dx^{1} … dx^{n}.

Fact. The change of variables formula ensures that this integral is welldefined on the intersection of any two charts.

To get a globally defined measure, we patch things together with a partition of unity: since M is second countable and locally compact, it follows that every open cover of M has a locally finite refinement. A partition of unity for a locally finite open cover {U_{α}} is a family of functions ϕ_{α} ∈ C^{∞}(M) such that

1. supp(ϕ_{α}) ⊂ U_{α};

2. ∑_{α} ϕ_{α} = 1.

Theorem. [6, Theorem 1.11] Any locally finite cover has a partition of unity.

Armed with this, we choose a locally finite cover of M by charts {(U_{α}, x_{α})}, a partition of unity {ϕ_{α}} for {U_{α}} and, for measurable f : M → ℝ, set

∫_{M} f dV = ∑_{α} ∫_{U_{α}} ϕ_{α}f dV.

Fact. This definition is independent of all choices.

2.3.2 The Divergence Theorem

Let X ∈ Γ(T M) have support in a chart (U, x).

∫_{M} div X dV = ∫_{U} \frac{1}{\sqrt{g}}∂_{i}(\sqrt{g}X_{i}) dV

= ∫_{x(U)}(∂_{i}\sqrt{g}X_{i}) ∘ x^{−1} dx^{1} … dx^{n}

= ∫_{x(U)} \frac{∂}{∂x^{i}} (\sqrt{g}X_{i}) ∘ x^{−1} dx^{1} … dx^{n} = 0.

A partition of unity argument immediately gives:

Divergence Theorem I. Any compactly supported vector field X on M has

∫_{M} div X dV = 0.

Just as in vector calculus, the divergence theorem quickly leads to Green’s formulae. Indeed, for f, h ∈ C^{∞}(M), X ∈ Γ(T M) one easily verifies:

div(fX) = f div X + g(grad f, X)

whence

div(f grad h) = fΔh + g(grad h, grad f)
Δ(fh) = fΔh + 2g(grad h, grad f) + hΔf.

The divergence theorem now gives us Green’s Formulae:

Theorem. For f, h ∈ C^{∞}(M) with at least one of f and h compactly supported:

∫_{M} hΔf dV = − ∫_{M} g(grad f, grad h) dV
∫_{M} hΔf dV = ∫_{M} fΔh dV.

2.3.3 Boundary terms

Supposed that M is oriented and that Ω ⊂ M is an open subset with smooth boundary ∂Ω. Thus ∂Ω is a smooth manifold with

1. a Riemannian metric inherited via (∂Ω)_{m} ⊂ M_{m}; 2. a Riemannian measure dA;

3. a unique outward-pointing normal unit vector field ν.

With these ingredients, one has:

Divergence Theorem II. Any compactly supported X on M has

∫_{Ω} div X dV = ∫_{∂Ω} g(X, ν) dA

and so Green’s Formulae:

Theorem. For f, h ∈ C^{∞}(M) with at least one of f and h compactly supported:

∫_{Ω} hΔf + ⟨grad f, grad h⟩ dV = ∫_{∂Ω} h⟨ν, grad f⟩ dA
∫_{Ω} hΔf − ∫_{Ω} fΔh dV = ∫_{∂Ω} h⟨ν, grad f⟩ dA − ∫_{∂Ω} f⟨ν, grad h⟩ dA

where we have written ⟨ , ⟩ for g( , ).

In particular

∫_{Ω} Δf dV = ∫_{∂Ω} νf dV.

3 Geodesics and curvature

In the classical geometry of Euclid, a starring role is played by the straight lines. Viewed as paths of shortest length between two points, these may be generalised to give a distinguished family of paths, the geodesics, on any Riemannian manifold. Geodesics provide a powerful tool to probe the geometry of Riemannian manifolds.

Notation. Let (M, g) be a Riemannian manifold. For ξ, η ∈ M_{m}, write

g(ξ, η) = ⟨ξ, η⟩, \sqrt{g(ξ, ξ)} = |ξ|.

3.1 (M, g) is a metric space

A piece-wise C^{1} path γ : [a, b] → M has length L(γ):

L(γ) = ∫_{a}^{b}|γ′(t)| dt.

Exercise. The length of a path is invariant under reparametrisation.

Recall that M is connected and so^{3} path-connected. For p, q ∈ M, set

d(p, q) = inf{L(γ) : γ : [a, b] → M is a path with γ(a) = p, γ(b) = q}.

One can prove:

• (M, d) is a metric space.

• The metric space topology coincides with the original topology on M.

The key points here are the definiteness of d and the assertion about the topologies. For this, it is enough to work in a precompact open subset of a chart U where one can prove the existence of K_{1}, K_{2} ∈ ℝ such that

K_{1} ∑_{1≤i≤n} ξ_{i}^{2} ≤ ∑_{i,j} g_{ij}ξ_{i}ξ_{j} ≤ K_{2} ∑_{1≤i≤n} ξ_{i}^{2}.

From this, one readily sees that, on such a subset, d is equivalent to the Euclidean metric on U.

3.2 Parallel vector fields and geodesics

Let c : I → M be a path. Recall the pull-back connection c^{−1}∇ on the space Γ(c^{−1}T M) of vector fields along c. This connection gives rise to a differential operator

∇_{t} : Γ(c^{−1}T M) → Γ(c^{−1}T M)

by

∇_{t}Y = (c^{−1}∇)_{∂_{1}}Y

where ∂_{1} is the coordinate vector field on I.

Note that since ∇ is metric, we have

⟨X, Y ⟩′ = ⟨∇_{t}X, Y ⟩ + ⟨X, ∇_{t}Y ⟩,

for X, Y ∈ Γ(c^{−1}T M).

Definition. X ∈ Γ(c^{−1}T M) is parallel if ∇_{t}X = 0.

The existence and uniqueness results for linear ODE give:

3Manifolds are locally path-connected!

Proposition. For c : [a, b] → M and U_{0} ∈ M_{c(a)}, there is unique parallel vector field U along c with

U(a) = U_{0}.

If Y_{1}, Y_{2} are parallel vector fields along c, then all ⟨Y_{i}, Y_{j}⟩ and, in particular, |Y_{i}| are constant.

Definition. γ : I → M is a geodesic if γ′ is parallel:

∇_{t}γ′ = 0.

It is easy to prove that, for a geodesic γ:

• |γ′| is constant.

• If γ is a geodesic, so is t ↦ γ(st) for s ∈ ℝ.

The existence and uniqueness results for ODE give:

1. For ξ ∈ M_{m}, there is a maximal open interval I_{ξ} ⊂ ℝ on which there is a unique geodesic γ_{ξ} : I_{ξ} → M such that

γ_{ξ}(0) = m
γ′_{ξ}(0) = ξ.

2. (t, ξ) ↦ γ_{ξ}(t) is a smooth map I_{ξ} × M_{m} → M.

3. γ_{sξ}(t) = γ_{ξ}(st).

Let us collect some examples:

1. M = ℝ^{n} with its canonical metric. The geodesic equation reduces to:

\frac{d^{2}γ}{dt^{2}} = 0

and we conclude that geodesics are straight lines.

2. M = S^{n} and ξ is a unit vector in M_{m} = m^{⊥}. Contemplate reflection in the 2-plane spanned by m and ξ: this induces a map Φ : S^{n} → S^{n} which preserves the metric and so ∇ also while it fixes m and ξ. Thus, if γ is a geodesic so is Φ ∘ γ and the uniqueness part of the ODE yoga forces Φ ∘ γ_{ξ} = γ_{ξ}. Otherwise said, γ_{ξ} lies in the plane spanned by m and ξ and so lies on a great circle.

To get further, recall that |γ′_{ξ}| = |ξ| = 1 which implies:

γ_{ξ}(t) = (cos t)m + (sin t)ξ.

A similar argument shows that the unique parallel vector field U along γ_{ξ} with U(0) = η ⊥ ξ is given by

U ≡ η.

3. M = D^{n} with the hyperbolic metric and ξ is a unit vector in M_{0} ≅ ℝ^{n}. Again, symmetry considerations force γ_{ξ} to lie on the straight line through 0 in the direction of ξ and then |γ′_{ξ}| = 1 gives:

γ_{ξ}(t) = (2 tanh t/2)ξ.

Similarly, the parallel vector field along γ_{ξ} with U(0) = η ⊥ ξ is given by

U(t) = \frac{1}{cosh^{2} t/2}η.

3.3 The exponential map

3.3.1 Normal coordinates

Set U_{m} = {ξ ∈ M_{m} : 1 ∈ I_{ξ}} and note that U_{m} is a star-shaped open neighbourhood of 0 ∈ M_{m}. We define the exponential map exp_{m} : U_{m} → M by

exp_{m}(ξ) = γ_{ξ}(1).

Observe that, for all t ∈ I_{ξ},

exp_{m}(tξ) = γ_{tξ}(1) = γ_{ξ}(t)

and differentiating this with respect to t at t = 0 gives

ξ = γ′_{ξ}(0) = (d exp_{m})_{0}(ξ)

so that (d exp_{m})_{0} = 1_{M_{m}}. Thus, by the inverse function theorem, exp_{m} is a local diffeomorphism whose inverse is a chart.

Indeed, if e_{1}, …, e_{n} is an orthonormal basis of M_{m}, we have normal coordinates x^{1}, …, x^{n} given by

x^{i} = ⟨(exp_{m})^{−1}, e_{i}⟩

for which

g_{ij}(m) = δ_{ij}
Γ_{ij}^{k}(m) = 0.

3.3.2 The Gauss Lemma

Let ξ, η ∈ M_{m} with |ξ| = 1 and ξ ⊥ η.

The Gauss Lemma says:

⟨(d exp_{m})_{tξ}η, γ′_{ξ}(t)⟩ = 0.

Thus γ_{ξ} intersects the image under exp_{m} of spheres in M_{m} orthogonally.

As an application, let us show that geodesics are locally length-minimising. For this, choose δ > 0 sufficiently small that

exp_{m} : B(0, δ) ⊂ M_{m} → M

is a diffeomorphism onto an open set U ⊂ M. Let c : I → U be a path from m to p ∈ U and let γ : I → U be the geodesic from m to p: thus γ is the image under exp_{m} of a radial line segment in B(0, δ).

Write

c(t) = exp_{m}(r(t)ξ(t))

with r : I → ℝ and ξ : I → S^{n} ⊂ M_{m}. Now

⟨c′(t), c′(t)⟩ = (r′)^{2} + r^{2}⟨(d exp_{m})_{rξ}ξ′, (d exp_{m})_{rξ}ξ′⟩ + 2rr′⟨(d exp_{m})_{rξ}ξ′, γ′_{ξ}⟩
= (r′)^{2} + r^{2}⟨(d exp_{m})_{rξ}ξ′, (d exp_{m})_{rξ}ξ′⟩

by the Gauss lemma (since ξ′ ⊥ ξ). In particular,

⟨c′(t), c′(t)⟩ ≥ (r′)^{2}.

Taking square roots and integrating gives:

L(c) ≥ ∫_{a}^{b}|r′| dt ≥ |∫_{a}^{b} r′ dt| = |r(b) − r(a)| = L(γ).

From this we conclude:

L(γ) = d(m, p)

and

B_{d}(m, δ) = exp_{m} B(0, δ).

Definition. A geodesic γ is minimising on [a, b] ⊂ I_{γ} if

L(γ_{|[a,b]}) = d(γ(a), γ(b)).

We have just seen that any geodesic is minimising on sufficiently small intervals.

3.3.3 The Hopf–Rinow Theorem

Definition. (M, g) is geodesically complete if I_{ξ} = ℝ, for any ξ ∈ ℝ.

This only depends on the metric space structure of (M, d):

Theorem (Hopf–Rinow). The following are equivalent:

1. (M, g) is geodesically complete.

2. For some m ∈ M, exp_{m} is a globally defined surjection M_{m} → M.

3. Closed, bounded subsets of (M, d) are compact.

4. (M, d) is a complete metric space.

In this situation, one can show that any two points of M can be joined by a minimising geodesic.

3.4 Sectional curvature

Let σ ⊂ M_{m} be a 2-plane with orthonormal basis ξ, η.

The sectional curvature K(σ) of σ is given by

K(σ) = ⟨R(ξ, η)ξ, η⟩.

Facts:

• This definition is independent of the choice of basis of σ.

• K determines the curvature tensor R.

Definition. (M, g) has constant curvature κ if K(σ) = κ for all 2-planes σ in T M.

In this case, we have

R(ξ, η)ζ = κ{⟨ξ, ζ⟩η − ⟨η, ζ⟩ξ}.

K is a function on the set (in fact manifold) G_{2}(T M) of all 2-planes in all tangent spaces M_{m} of M. A diffeomorphism Φ : M → M induces dΦ : T M → T M which is a linear isomorphism on each tangent space and so gives a mapping Φ̂ : G_{2}(T M) → G_{2}(T M). Suppose now that Φ is an isometry:

⟨dΦ_{m}(ξ), dΦ_{m}(η)⟩ = ⟨ξ, η⟩,

for all ξ, η ∈ M_{m}, m ∈ M. Since an isometry preserves the metric, it will preserve anything built out of the metric such as the Levi–Civita connection and its curvature. In particular, we have

K ∘ Φ̂ = K.

It is not too difficult to show that, for our canonical examples, the group of all isometries acts transitively on G_{2}(T M) so that K is constant. Thus we arrive at the following examples of manifolds of constant curvature:

1. ℝ^{n}.

2. S^{n}(r).

3. D^{n}(ρ) with metric

g_{ij} = \frac{4δ_{ij}}{(1 − |z|^{2}/ρ^{2})^{2}} .

It can be shown that these exhaust all complete, simply-connected possibilities.

3.5 Jacobi fields

Definition. Let γ : I → M be a unit speed geodesic. Say Y ∈ Γ(γ^{−1}T M) is a Jacobi field along γ if

∇_{t}^{2}Y + R(γ′, Y)γ′ = 0.

Once again we wheel out the existence and uniqueness theorems for ODE which tell us:

Proposition. For Y_{0}, Y_{1} ∈ M_{γ(0)}, there is a unique Jacobi field Y with

Y (0) = Y_{0}
(∇_{t}Y)(0) = Y_{1}

Jacobi fields are infinitesimal variations of γ through a family of geodesics. Indeed, suppose that h : I × (−ϵ, ϵ) → M is a variation of geodesics: that is, each γ_{s} : t → h(t, s) is a geodesic. Set γ = γ_{0} and let

Y = \frac{∂h}{∂s}|_{s=0} ∈ Γ(γ^{−1}T M).

Let ∂_{t} and ∂_{s} denote the coordinate vector fields on I × (−ϵ, ϵ) and set D = h^{−1}∇. Since each γ_{s} is a geodesic, we have

D_{∂_{t}} \frac{∂h}{∂t} = 0

whence

D_{∂_{s}}D_{∂_{t}} \frac{∂h}{∂t} = 0.

The definition of the curvature tensor, along with the fact that [∂_{s}, ∂_{t}] = 0, allows us to write

0 = D_{∂_{s}}D_{∂_{t}} \frac{∂h}{∂t} = D_{∂_{t}}D_{∂_{s}} \frac{∂h}{∂t} + R(\frac{∂h}{∂t}, \frac{∂h}{∂s})\frac{∂h}{∂t}.

Moreover, it follows from the fact that ∇ is torsion-free that

D_{∂_{s}} \frac{∂h}{∂t} = D_{∂_{t}} \frac{∂h}{∂s}

so that

0 = (D_{∂_{t}})^{2} \frac{∂h}{∂s} + R(\frac{∂h}{∂t}, \frac{∂h}{∂s})\frac{∂h}{∂t}.

Setting s = 0, this last becomes

(∇_{t})^{2}Y + R(γ′, Y)γ′ = 0.

Fact. All Jacobi fields arise this way.

Let us contemplate an example which will compute for us the (constant) value of K for hyperbolic space: let (D^{n}, g) be hyperbolic space and consider a path ξ : (−ϵ, ϵ) → S^{n−1} ⊂ D_{0} with ξ′(0) = η ⊥ ξ(0).

We set h(t, s) = γ_{ξ(s)}(t) = (2 tanh t/2)ξ(s)—a variation of geodesics through 0. We then have a Jacobi field Y along γ = γ_{ξ(0)}:

Y (t) = \frac{∂h}{∂s}|_{s=0} = 2(tanh t/2)η
= sinh t(η/ cosh^{2} t/2)
= sinh tU(t)

where U is a unit length parallel vector field along γ.

We therefore have:

(∇_{t})^{2}Y = sinh′′ tU(t) = sinh tU(t)

whence

U + R(γ′, U)γ′ = 0.

Take an inner product with U to get

K(γ′ ∧ U) = −1

and so conclude that (D^{n}, g) has constant curvature −1.

The same argument (that is, differentiate the image under exp_{m} of a family of straight lines through the origin) computes Jacobi fields in normal coordinates:

Theorem. For ξ ∈ M_{m}, the Jacobi field Y along γ_{ξ} with

Y (0) = 0
(∇_{t}Y)(0) = η ∈ M_{m}

is given by

Y (t) = (d exp)_{tξ}tη.

3.6 Conjugate points and the Cartan–Hadamard theorem

Let ξ ∈ M_{p} and let γ = γ_{ξ} : I_{ξ} → ℝ. We say that q = γ(t_{1}) is conjugate to p along γ if there is a non-zero Jacobi field Y with

Y (0) = Y (t_{1}) = 0.

In view of the theorem just stated, this happens exactly when (d exp_{p})_{t_{1}ξ} is singular.

Theorem (Cartan–Hadamard). If (M, g) is complete and K ≤ 0 then no p ∈ M has conjugate points.

Proof. Suppose that Y is a Jacobi field along some geodesic γ with Y (0) = Y (t_{1}) = 0. Then

0 = ∫_{0}^{t_{1}}⟨∇_{t}^{2}Y + R(γ′, Y)γ′, Y ⟩ dt
= − ∫_{0}^{t_{1}}|∇_{t}T |^{2} dt + ∫_{0}^{t_{1}} K(γ′ ∧ Y)|Y |^{2} dt

where we have integrated by parts and used Y (0) = Y (t_{1}) = 0 to kill the boundary term. Now both summands in this last equation are non-negative and so must vanish. In particular,

∇_{t}Y = 0

so that Y is parallel whence |Y | is constant giving eventually Y ≡ 0.

From this we see that, under the hypotheses of the theorem, each exp_{m} is a local diffeomorphism and, with a little more work, one can show that exp_{m} : M_{m} → M is a covering map. Thus:

Corollary. If (M, g) is complete and K ≤ 0 then

1. if π_{1}(M) = 1 then M is diffeomorphic to ℝ^{n}.

2. In any case, the universal cover of M is diffeomorphic to ℝ^{n} whence π_{k}(M) = 1 for all k ≥ 2.

Analysis of this kind is the starting point of one of the central themes of modern Riemannian geometry: the interplay between curvature and topology.

4 The Bishop volume comparison theorem

Our aim is to prove a Real Live Theorem in Riemannian geometry: the theorem is of considerable interest in its own right and proving it will exercise everything we have studied in these notes.

We begin by collecting some ingredients.

4.1 Ingredients

4.1.1 Ricci curvature

Definition. The Ricci tensor at m ∈ M is the bilinear map Ric : M_{m} × M_{m} → ℝ given by

Ric(ξ, η) = trace(ζ ↦ R(ξ, ζ)η)

= ∑_{i}⟨R(ξ, e_{i})η, e_{i}⟩

where e_{1}, …, e_{n} is an orthonormal basis of M_{m}.

Exercise. The Ricci tensor is symmetric: Ric(ξ, η) = Ric(η, ξ).

Example. If (M, g) has dimension n and constant curvature κ then

Ric = (n − 1)κg.

The Ricci tensor, being only bilinear, is much easier to think about than the curvature tensor. On the other hand, being only an average of sectional curvatures, conditions of the Ricci tensor say much less about the topology of the underlying manifold. For example, here is an amazing theorem of Lohkamp [4]:

Theorem. Any manifold of dimension at least 3 admits a complete metric with Ric < 0 (that is Ric is neagtive definite).

4.1.2 Cut locus

Henceforth, we will take M to be complete of dimension n.

For ξ ∈ M_{m} with |ξ| = 1, define c(ξ) ∈ ℝ^{+} ∪ {∞} by

c(ξ) = sup{t: γ_{ξ}|_{[0,t]} is minimising}
= sup{t: d(m, γ_{ξ}(t)) = t}.

The cut locus C_{m} of m is given by

C_{m} = exp_{m}{c(ξ)ξ: ξ ∈ S^{n−1} ⊂ M_{m}, c(ξ) < ∞}

while D_{m} = {tξ: ξ ∈ S^{n−1} ⊂ M_{m}, t ∈ [0, c(ξ))} and

D_{m} = exp_{m} D_{m}.

We have:

• M = D_{m} ∪ C_{m} is a disjoint union.

• exp_{m} : D_{m} → D_{m} is a diffeomorphism.

• ∫_{C_{m}} dV = 0.

These facts have practical consequences for integration on M: for f : M → ℝ integrable,

∫_{M} f dV = ∫_{D_{m}} f(exp(x))\sqrt{g} dx^{1} … dx^{n}
= ∫_{S^{n−1}} ∫_{0}^{c(ξ)} f(exp(rξ))a(r, ξ) drdξ

where x^{1}, …, x^{n} are orthonormal coordinates on D_{m} and dξ is Lebesgue measure on S^{n−1} ⊂ M_{m}.

Example. For κ ∈ ℝ, let S_{κ} : ℝ → ℝ solve

S′′_{κ} + κS_{κ} = 0
S_{κ}(0) = 0, S′_{κ}(0) = 1

Then, if (M, g) has constant curvature κ,

a(r, ξ) = S_{κ}^{n−1}(r).

4.2 Bishop’s Theorem

4.2.1 Manifesto

Fix κ ∈ ℝ and m ∈ M_{m}.

Let V (m, r) denote the volume of B_{d}(m, r) ⊂ M and V_{κ}(r) the volume of a radius r ball in a complete simply-connected n-dimensional space of constant curvature κ.

Suppose that Ric(ξ, ξ) ≥ (n − 1)κg(ξ, ξ) for all ξ ∈ T M. For each ξ ∈ M_{m} of unit length, define a_{ξ} : (0, c(ξ)) → ℝ by

a_{ξ}(t) = a(t, ξ).

We will prove that

\frac{a′_{ξ}}{a_{ξ}} ≤ (n − 1)\frac{S′_{κ}}{S_{κ}} .

As a consequence, we will see that

V (m, r) ≤ V_{κ}(r)

and even that V (m, r)/V_{κ}(r) is decreasing with respect to r.

4.2.2 Laplacian of the distance function

Our strategy will be to identify the radial logarithmic derivative of a with the Laplacian of the distance from m. We will then be able to apply a formula of Lichnerowicz to derive a differential inequality for a_{ξ}.

So view r as a function on M:

r(x) = d(m, x).

Then

Proposition. a^{−1}∂a/∂r = Δr ∘ exp_{m}.

Here is a fast^{4} proof stolen from [3]: for U ⊂ S^{n−1} ⊂ M_{m} and [t, t + ϵ] such that

Ω_{t,ϵ} = {exp_{m}(rξ): r ∈ [t, t + ϵ], ξ ∈ U} ⊂ D_{m}

4Isaac Chavel rightly objects that this proof is all a bit too slick. See his contribution to this volume for a more down to earth proof.

we have:

∫_{Ω_{t,ϵ}} Δr dV = ∫_{[t,t+ϵ]×U}(Δ ∘ exp_{m})a drdξ.

However, the divergence theorem gives

∫_{Ω_{t,ϵ}} Δ dV = ∫_{∂Ω_{t,ϵ}}⟨grad r, ν⟩ dA = ∫_{U} a(t + ϵ) dξ − ∫_{U} a(t) dξ

= ∫_{U} ∫_{t}^{t+ϵ} \frac{∂a}{∂r}(r, ξ) drdξ.

Here we have used that ⟨grad r, ν⟩ = νr = 1 along the spherical parts of ∂Ω_{t,ϵ} and vanishes along the radial parts.

Thus

∫_{[t,t+ϵ]×U}(Δ ∘ exp_{m})a drdξ = ∫_{[t,t+ϵ]×U} \frac{∂a}{∂r}(r, ξ) drdξ

and, since t, ϵ and U were arbitrary, we get

a(Δ ∘ exp_{m}) = \frac{∂a}{∂r}

as required.

4.2.3 Lichnerowicz’ formula

For X ∈ Γ(T M), define |∇X|^{2} by

|∇X|^{2}(m) = ∑_{i}|∇_{e_{i}}X|^{2}

where e_{1}, …, e_{n} is an orthonormal basis of M_{m}—this is independent of choices.

We now have

Lichnernowicz’ Formula. Let f : M → ℝ then

\frac{1}{2}Δ|grad f|^{2} = |∇ grad f|^{2} + ⟨grad Δf, grad f⟩ + Ric(grad f, grad f).

The proof of this is an exercise (really!) but here are some hints to get you started: the basic identity

XY f − Y Xf = [X, Y]

along with the fact that ∇ is metric and torsion-free gives:

⟨∇_{X} grad f, Y ⟩ = ⟨∇_{Y} grad f, X⟩

from which you can deduce that

\frac{1}{2} grad|grad f|^{2} = ∇_{grad f} grad f

whence

\frac{1}{2}Δ|grad f|^{2} = div ∇_{grad f} grad f

= ∑_{i}⟨∇_{e_{i}}∇_{grad f} grad f, e_{i}⟩.

Now make repeated use of the metric property of ∇ and use the definition of R to change the order of the differentiations …

As an application, put f = r. Thanks to the Gauss lemma, grad f = ∂_{r} so that |grad f| = 1 and the Lichnerowicz formula reads:

0 = |∇ grad r|^{2} + ∂_{r}Δr + Ric(∂_{r}, ∂_{r}).  (4.1)

On the image of γ_{ξ}, we have

∂_{r}Δr = (a′_{ξ}/a_{ξ})′ = a′′_{ξ}/a_{ξ} − (Δr)^{2}

and plugging this into (4.1) gives

0 = a′′_{ξ}/a_{ξ} − (Δr)^{2} + |∇ grad r|^{2} + Ric(∂_{r}, ∂_{r})

or, defining b by b^{n−1} = a_{ξ} so that (n − 1)b′/b = a′_{ξ}/a_{ξ},

(n − 1)b′′/b + Ric(∂_{r}, ∂_{r}) = −(|∇ grad r|^{2} − \frac{1}{n − 1}(Δr)^{2}).  (4.2)

4.2.4 Estimates and comparisons

We now show that the right hand side of (4.2) has a sign: choose an orthonormal basis e_{1}, …, e_{n} of M_{γ_{ξ}(t)} with e_{1} = ∂_{r}. Then

Δr = ∑⟨∇_{e_{i}} grad r, e_{i}⟩

= ∑_{i≥2}⟨∇_{e_{i}} grad r, e_{i}⟩

since ∇_{∂_{r}} grad r = ∇_{t}γ′_{ξ} = 0.

Two applications of the Cauchy–Schwarz inequality give

(Δr)^{2} ≤ (∑_{i≥2}|∇_{e_{i}} grad r|)^{2} ≤ (n − 1) ∑_{i≥2}|∇_{e_{i}} grad r|^{2}

so that

|∇ grad r|^{2} − \frac{1}{n − 1}(Δr)^{2} ≥ 0.

Thus (4.2) gives

(n − 1)b′′/b + Ric(∂_{r}, ∂_{r}) ≤ 0

and, under the hypotheses of Bishop’s theorem, we have

b′′/b ≤ −κ.

We now make a simple comparison argument: b > 0 on (0, c(ξ)) so we have

b′′ + κb ≤ 0
b(0) = 0, b′(0) = 1.

On the other hand, set b̄ = S_{κ} so that

b̄′′ + κb̄ = 0
b̄(0) = 0, b̄′(0) = 1

We now see that, so long as b̄ ≥ 0, we have

b̄b′′ − b̄′′b ≤ 0

or, equivalently,

(b′b̄ − b̄′b)′ ≤ 0.

In view of the initial conditions, we conclude:

b′b̄ − b̄′b ≤ 0.  (4.3)

Let us pause to observe that at the first zero of b̄ (if there is one), b̄′ < 0 so that, by (4.3), b ≤ 0 also. Since b > 0 on (0, c(ξ)), we deduce that b̄ > 0 there also^{5}.

We therefore conclude from (4.3) that on (0, c(ξ)) we have

b′/b ≤ b̄′/b̄,

or, equivalently,

a′_{ξ}/a_{ξ} ≤ (n − 1)S′_{κ}/S_{κ}.  (4.4)

5For κ > 0, this reasoning puts an upper bound on the length of (0, c(ξ)) and thus, eventually, on the diameter of M. This leads to a proof of the Bonnet–Myers theorem.

4.2.5 Baking the cake

Equation (4.4) reads

ln(a_{ξ}/S_{κ}^{n−1})′ ≤ 0

so that, a_{ξ}/S_{κ}^{n−1} is decreasing and, in view of the initial conditions,

a_{ξ} ≤ S_{κ}^{n−1}.

Thus:

V (m, r) = ∫_{S^{n−1}} ∫_{0}^{min(c(ξ),r)} a_{ξ} drdξ
≤ ∫_{S^{n−1}} ∫_{0}^{min(c(ξ),r)} S_{κ}^{n−1} drdξ = V_{κ}(r).

This is Bishop’s theorem.

Our final statement is due to Gromov [2] and is a consequence of a simple lemma:

Lemma ([2]). If f, g > 0 with f/g decreasing then

∫_{0}^{r} f/ ∫_{0}^{r} g

is decreasing also.

With this in hand, we see that, for r_{1} < r_{2},

∫_{0}^{r_{1}} a_{ξ} dr/ ∫_{0}^{r_{1}} S_{κ}^{n−1} dr ≤ ∫_{0}^{r_{2}} a_{ξ} dr/ ∫_{0}^{r_{2}} S_{κ}^{n−1} dr.

Integrating this over S^{n−1}, noting that the denominators are independent of ξ, gives finally that V (m, r)/V_{κ}(r) is decreasing.

References

[1] I. Chavel, Riemannian geometry—a modern introduction, Cambridge University Press, Cambridge, 1993. 1

[2] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), no. 1, 15–53. 28, 28 [3] S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Springer-

Verlag, Berlin, 1987. 1, 1, 24

[4] J. Lohkamp, Metrics of negative Ricci curvature, Ann. of Math. (2) 140 (1994), no. 3, 655–683. 22

[5] M. Spivak, Calculus on manifolds. A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965. 1

[6] F.W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, Ill.-London, 1971. 1, 2, 11