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ELEMENTARY RIEMANNIAN GEOMETRY
VISHWAMBHAR PATI
Abstract. In these lectures, we cover some basic material on Riemannian Geometry.
1. Advanced Calculus
1.1. Derivatives. Insinglevariablecalculus, onesaysthatareal-valuedfunctionf : (c,d) → Risdifferentiable
at a ∈ (c,d) if the limit f(a+h)−f(a)
lim
h→0 h
exists. If it does, one calls this limit L the derivative of f at a, and denotes it by L = df (a) or f′(a).
dx
In order to generalise this to a real-valued function of several variables, i.e. a function (or map) f : U → R,
where U ⊂ Rn is an open set, one cannot blindly carry over the above one-variable definition, because one
would have to divide the scalar f(a + h) − f(a) by the vector h ∈ Rn. But one may recast the one-variable
definition above in the following form:
f(a+h)−f(a)
lim −L =0
which is the same as saying: h→0
h
lim kf(a+h)−f(a)−Lhk =0
khk→0 khk
This last fomulation easily generalises to functions of several variables. One merely has to note that the
scalar L must now be replaced by some map that can be applied to the vector h ∈ Rn to yield a scalar. Also,
since multiplication by a scalar is a linear map from R to R, it is reasonable to require L : Rn → R to be a
linear map. This motivates the following:
Definition 1.1.1 (Differentiability). Let U ⊂ Rn be an open set, and f : U → Rm be a map. One says that
f is differentiable at a ∈ U if there exists a linear map L : Rn → Rm satisfying:
lim kf(a+h)−f(a)−L:hk =0
khk→0 khk
The linear map L is called the derivative of f at a and denoted by L = Df(a). If f is differentiable at all
points a ∈ U, we say f is differentiable on U.
Remark 1.1.2. Another reformulation of the definition above is: there exists a linear map L such that for h
small enough,:
f(a+h)=f(a)+L:h+g(h)
where g(h) = o(khk), viz. g(h)
lim khk =0
khk→0
This formulation says that there is a linear approximation L to the map f, in the sense that the discrepancy
between f(a+h)−f(a) and L:h is of “order strictly higher than 1” in khk, for small h.
1
2 VISHWAMBHAR PATI
Exercise 1.1.3.
(i): (Derivative is well-defined) Show that if a linear map L exists, in accordance with the Definition 1.1.1
above, then it is unique.
(ii): Show that f is differentiable at a implies that f is continuous at a.
(iii): (The Chain Rule) If f : U → Rm and g : V → Rl where U ⊂ Rn is open, V ⊂ Rm is open and contains
f(U), f is differentiable at a ∈ U, and g is differentiable at f(a) ∈ V , then the composite g ◦f : U → Rl
is differentiable at a, and has the derivative Dg(f(a)) ◦ Df(a). (Hint: Use the Remark 3.3.12).
(iv): Show that f : U → Rm is differentiable at a ∈ U iff each of the component functions f (1 ≤ i ≤ m)
i
of f is differentiable at a, and that
Df (a)
1
Df (a)
2
:
Df(a) =
:
Df:(a)
m
(v): If L : Rn → Rm is a linear map, then its derivative is L at all points of Rn (the best linear approximation
of a linear map is itself). More generally, for an affine map f : Rn → Rm, (that is, f(x) = Lx+b, where
L:Rn→Rmisalinear map, and b∈Rm is some fixed vector) the derivative Df(a) = L for all a ∈ Rn.
(vi): Show that if f : U → Rm and g : U → Rm are differentiable at a ∈ U, U ⊂ Rn an open set, then so is
αf +βg for fixed α,β ∈ R. When m = 1, then show that the map fg (defined by fg(x) = f(x)g(x)) is
−1
also differentiable at a. If f(x) 6= 0 for all x ∈ U, show that x 7→ (f(x)) is also differentiable at a. Thus
any polynomial function on Rn is differentiable at all points in Rn, and a rational function (i.e. f = P=Q,
where P,Q are two polynomial functions) is differentiable at all points of the open set Rn \Z(Q), where
−1
Z(Q) is the closed zero-set Q (0) of Q.
(vii): One can identify the vector space M(n,R) of n × n real matrices with the euclidean space Rn2. Let
GL(n,R) ⊂ M(n,R) be the open set of all real nonsingular matrices. Show that the bijective map of
GL(n,R) to itself defined by A 7→ A−1 is differentiable at all points in GL(n,R). (Use Cramer’s formula
for the inverse of a matrix).
1.2. Directional and partial derivatives.
Definition 1.2.1 (Directional and partial derivatives). Let U ⊂ Rn, and f : U → Rm be differentiable at a
point a ∈ U. Let v ∈ Rn be some vector. The directional derivative of f along v is the quantity:
lim f(a+tv)−f(a)
t→0 t
This quantity exists, and is easily shown to be Df(a)v (Exercise).
n n
Let {e } be the standard basis of R . The directional derivatives along these directions e are called the
i i=1 i
partial derivatives of f. If we write f = (f ,::,f ) in terms of its components, and denote:
1 m
∂f
i (a) := Df (a)(e )
∂x i j
j
then with respect to the standard bases of Rn and Rm respectively, the linear map Df(a) can be represented
as the matrix:
ELEMENTARY RIEMANNIAN GEOMETRY 3
Df (a)
1 ∂f ∂f ∂f
Df (a) 1 (a) 1 (a) ::: 1 (a)
2 ∂x ∂x ∂x
1 2 n
: : : ::: :
Df(a) = =
: : : ::: :
∂f ∂f ∂f
: m(a) m(a) ::: m(a)
∂x ∂x ∂x
Df (a) 1 2 n
m
The m×nmatrix on the right is called the Jacobian matrix of f. Note that a matrix representation for Df(a)
can be done with respect to any bases of Rn and Rm respectively, but the Jacobian matrix is the representation
of Df(a) with respect to the standard bases.
Remark 1.2.2 (Caution). Note that for a function f as above, differentiable or not, it is possible to define
the partial derivative:
∂f f(a+te )−f(a)
(a) := lim j
∂x t→0 t
j
if it exists. Further, it is possible for all the partial derivatives ∂f (a) of a function f (indeed, even all directional
∂xj
derivatives) to exist at a point a without the function f being differentiable at a. (See Exercise 1.3.3 below).
1.3. Higher derivatives, smooth functions and maps. We note that the space of linear maps from Rn
to Rm is itself a linear vector space, of dimension mn. If we denote this vector space by hom (Rn,Rm), then
R
by choice of bases {e } and {f } of Rn and Rm respectively, we can identify this vector space with the vector
i j
space of m×n real matrices, which is isomorphic to Rmn.
Definition 1.3.1 (Cr-maps). Let U ⊂ Rn be open, and f : U → Rm be a map. If f is differentiable on U
and the map:
Df : U → hom (Rn,Rm)
R
a 7→ Df(a)
is continuous, then we say f is C1. More generally, we inductively define f to be Cr if the map Df above is
Cr−1. If a function is Cr for all r ≥ 1, then we say f is C∞ or smooth. By convention, a C0 map means a
continuous map.
i i i
Exercise 1.3.2. Show that f is Cr implies that all its mixed partial derivatives Dαf := ∂ 1∂ 2::∂ n f exist
i i1 i2 in i
∂x ∂x ::∂xn
P 1 2
and are continuous for all 1 ≤ i ≤ m and all multi-indices α := (i ,::,i ) with |α| := i ≤r. Conversely,
1 n j j
show that if all these mixed partials exist and are continuous everywhere, then f is Cr.
Exercise 1.3.3. Consider the function
f : R2 → R
(x,y) 7→ x|y| (x,y) 6= (0,0)
2 2 1=2
(x +y )
(0,0) 7→ 0
Show that the function above is not differentiable at (0,0), even though both partial derivatives exist at the
origin. In fact, show that the restriction of f to every line passing through the origin is smooth, and hence all
directional derivatives of f exist at (0,0).
4 VISHWAMBHAR PATI
1.4. Diffeomorphisms.
Definition 1.4.1 (Diffeomorphism). Let 1 ≤ r ≤ ∞. A Cr map f : U → V, where U and V are open subsets
of some euclidean spaces, is called a Cr-diffeomorphism if there exists a Cr map g : V → U which satisfies
g ◦ f = Id , f ◦g = Id . A C∞-diffeomorphism is called a smooth diffeomorphism. For r = 0, the definition
U V
also makes sense, though it is more customary to call a C0-diffeomorphism a homeomorphism.
Remark 1.4.2. The Jacobian of a Cr-diffeomorphism f : U → V is pointwise invertible, as a linear map,
for r ≥ 1. For, by the chain rule of Exercise 1.1.3 (iii), it follows that the linear maps Df(a) and Dg(f(a))
are inverses of each other for each a ∈ U. Thus U and V have to be subsets of euclidean spaces of the same
dimension if they are Cr-diffeomorphic, for r ≥ 1. (For r = 0, the result is still true and is much harder to
prove. It is called the Brouwer Domain Invariance Theorem, and requires the use of homology theory. See e.g.
E. H. Spanier’s book Algebraic Topology for a proof).
Exercise 1.4.3. Let f be a Cr-map which is a C1- diffeomorphism, for r ≥ 1. Then f is a Cr-diffeomorphism
(Use (vii) of the Exercise 1.1.3). Show by an example that this conclusion is false if C1 above is replaced by
C0. That is, give an example of a C1-map which is a homeomorphism but not a C1-diffeomorphism.
Definition 1.4.4 (Local diffeomorphism). Let f : U → Rn be a Cr map, where U is an open set in Rn. We
say that f is a Cr-local diffeomorphism at a ∈ U if there is a neighbourhood W of a such that f : W →f(W)
r r |W
is a C - diffeomorphism. Note that a map which is a C -local diffeomorphism at every point of U will also
have invertible Jacobian at every point of U, for r ≥ 1.
Example 1.4.5 (A local diffeomorphism which is not a diffeomorphism). Consider the map:
f : R2 → R2
x x
(x,y) 7→ (e cos y,e sin y)
z
This is nothing but the map z 7→ e from C to C, written out in long-hand. It is clearly not a diffeomorphism
because the points (0,2nπ), n ∈ Z all map to (1,0). However, it is not difficult to show that f restricted to
any open strip R × (b,b + 2π) is a diffeomorphism. f maps this open strip diffeomorphically to the half-slit
plane, i.e. R2 minus the half-ray {(λcos b,λsin b) : λ ≥ 0}.
Example 1.4.6 (Polar Coordinates). There is also the (related) smooth map which is a local diffeomorphism:
f : (0,∞)×R → R2
(r,θ) 7→ (rcos θ,rsin θ)
Over any open strip (0,∞)×(α,α+2π), it is a diffeomorphism on to a half slit plane. Note that the smooth
t
diffeomorphism t → e taking R to (0,∞) converts this example to the previous one.
Exercise 1.4.7. Prove that a smooth bijection f : U → V of open subsets of Rn is a smooth diffeomorphism
iff it is a C1-local diffeomorphism at each point of U.
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