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BASIC GEOMETRY OF SUBMANIFOLDS
WERNERBALLMANN
Contents
Introduction 2
0.1. Notation 2
1. Intrinsic Geometry 3
1.1. First fundamental form 4
1.2. Intrinsic Distance 6
1.3. First Variation of Arc Length and Geodesics 7
1.4. Parallel vector fields 12
2. Extrinsic Geometry 14
2.1. Hypersurfaces 17
3. More examples and some exercises 21
Acknowledgments 25
References 25
Date: Last update: 30.11.02.
1
2 WERNERBALLMANN
Introduction
n
I discuss the geometry of submanifolds in R , which motivated most of the
results and techniques of present day differential geometry. This is not a historical
account of differential geometry. With today’s point of view I explain some of the
main insights into differential geometry before the time of Gauss and Riemann.
There are two aspects of the geometry of submanifolds, intrinsic geometry and
extrinsic geometry. In our setting, intrinsic differential geometry describes the
geometry inside the submanifolds — the only role of the ambient space Rn is to
induce a way of measuring angles and lengths of geometric objects contained in
the submanifolds. Extrinsic geometry deals with the shape of submanifolds as
subsets of the ambient space.
All considerations are local and the arguments go through for immersions as
well. Since many of the elementary examples come as immersions, the setting in
n
this chapter is the following: M is a manifold of dimension m, and f : M → R
is an immersion. The data is the pair (M,f).
n
I assume that the reader is familiar with the theory of curves in R . I also
assume some familiarity with the theory of manifolds. The reader who is not on
good terms with manifolds yet is advised to think of them as open subsets or
submanifolds of Euclidean spaces.
m
0.1. Notation. If U ⊂ R is open, V is a real vector space (of finite dimension),
and ϕ : U → V is a smooth function, then the partial derivative of ϕ with respect
to xi is denoted in the following different ways,
ϕ =ϕ = ∂ϕ =dϕ· ∂ .
i x
i ∂xi ∂xi
Analogous notation will be used for higher partial derivatives. There are other
objects with indices, where the indices have a different meaning. But it seems
that there is no danger of confusion.
We let F(M) and V(M) be the spaces of smooth real valued functions and
smooth vector fields on M, respectively. Recall that tangent vectors of M act
as derivations on smooth maps with values in vector spaces, ϕ : M → V. For
X∈V(M), we use the notations Xf = df ·X for the induced smooth function
M∋p7→X(p)(f)∈V.
BASIC GEOMETRY OF SUBMANIFOLDS 3
1. Intrinsic Geometry
Theintrinsic geometry of M with respect to the given map f is concerned with
the measurements of objects inside M. The only way in which Euclidean space
Rn enters is through the restriction of the inner product to the tangent spaces of
(M,f).
Definition 1.1. For p ∈ M, the tangent space Tpf and the normal space Npf of
(M,f) are the linear subspaces of Rn defined by
T f := imdf(p) and N f :=[imdf(p)]⊥.
p p
Since f is an immersion, df(p) : T M → T f is an isomorphism for all p ∈ M.
p p
In the case where M is a submanifold and f is the inclusion, df(p) is the usual
identification of T M with a linear subspace of Rn. In this case we write T M and
p p
N M instead of T f and N f. In the general case of an immersion, we also think
p p p
of df(p) as a natural identification of TpM with Tpf. For all p ∈ M, the tuple
(f (p),...,f (p)) of partial derivatives of f is a basis of T f. The codimension
1 m p
of (M,f) is n−m, it is equal to the dimension of Npf.
m m+1 2 2
Example 1.2. Let r > 0 and S ={x∈R | kxk = r } be the round sphere
r
of dimension m and radius r. Then Sm is a submanifold of Rm+1. Here f is the
r
inclusion, T Sm = {y ∈ Rm+1 | hx,yi = 0}, and N M = R·x.
x x
n T N
Let W ⊂ M. For a map X : W → R and p ∈ W, let X(p) = X (p)+X (p)
be the decomposition according to the splitting Rn = T f + N f. We call XT
p p
the tangential and XN the normal part of X. We start with a technical lemma,
which we want to have out of the way.
n T
Lemma 1.3. 1) If W ⊂ M is open, and X : W → R is smooth, then X and
XN are smooth.
2) For any point p ∈ M there are an open neighborhood W of p in M and
smooth maps X : W → Rn, 1 ≤ i ≤ n, such that (X (q),...,X (q)) is an
i 1 m
orthonormal basis of T f and (X (q),...,X (q)) an orthonormal basis of N f,
q m+1 m q
for all q ∈ U.
Proof. Let U → U′ beacoordinatechartofM aboutp. Choosev , . . . , v ∈Rn
m+1 n
such that (f (p),...,f (p),v , . . . , v ) is a basis of Rn. By continuity, there is
1 m m+1 n
an open neighborhood W ⊂ U of p such that (f (q),...,f (q),v , . . . , v ) is
1 m m+1 n
n
still a basis of R . Now apply Gram-Schmidt orthonormalization to this basis to
obtain an orthonormal basis X (q),...,X (q) of Rn. By the formulas defining
1 n
Gram-Schmidt orthonormalization, we see that the maps Xi are smooth.
Gram-Schmidtorthonormalization has the characteristic property that it keeps
linear hulls of any initial subsequence. In particular, the hull of X (q),...,X (q)
1 m
is the same as that of f (q),...,f (q). Hence (X (q),...,X (q)) is an orthonor-
1 m 1 n
mal basis of T f. It follows that (X (q),...,X (q)) is an orthonormal basis
q m+1 m
of Nqf. This proves Assertion 2). Assertion 1) is a direct consequence of 2).
4 WERNERBALLMANN
1.1. First fundamental form. In this section we discuss length measurements,
that is, we are concerned with the lengths of curves in M. Let c : [a,b] → M be
a piecewise smooth curve, that is, there is a subdivision
a = t < t < ... < t = b,
0 1 k
such that c|[t , t ] is smooth, 1 ≤ i ≤ k. We define the length L(c) of c by
i−1 i Z Z
b b
(1.1) L(c) = k(f ◦ c)′kdt = kdf(c(t)) · c′(t)kdt.
a a
Werecall that the length of c is invariant under reparameterizations of c.
The inner product and the norm of Euclidean space enter the definition of
length only via their restriction to the tangent spaces T f. To emphasize that
c(t)
the ambient space does not enter in any other way, we define a family g , p ∈ M,
p
of inner products on the respective tangent spaces of M as follows.
Definition 1.4. Thefirst fundamental formg of(M,f)atpistheinnerproduct
p
on T M given by
p
gp(v,w) := hdf(p) · v,df(p) · wi, v,w ∈ TpM.
Note that the formula defines an inner product indeed since f is an immersion.
Wedenotethefirstfundamentalformg alsobyh.,.i . Becausethereisnodanger
p p
of confusion, we will mostly delete the index p. In terms of the first fundamental
form, the length of the curve c above is given by
(1.2) L(c) = Z bkc′(t)kdt,
a
wherek.k = k.k denotesthenormassociatedtog . Bydefinition, L(c) = L(f◦c).
p p
So far, the first fundamental form looks a bit abstract and the question arises
how we can handle this object. The most straightforward way is by coordinate
charts: Let x : U → U′ be a coordinate chart of M. For all p ∈ U,
∂ (p),..., ∂ (p)
∂x1 ∂xm
is a basis of T M, and the coefficients g =g (p) of the first fundamental form
p ij ij
with respect to this basis are given by
∂ ∂
(1.3) g (p) := df(p)· (p),df(p) · (p) = hf (p),f (p)i.
ij ∂xi ∂xj i j
Since U ∋ p 7→ f (p) is smooth for all i, it follows that the coefficients g are
i ij
smooth functions on U.
Examples 1.5. 1) Graphs: Let W ⊂ Rm be an open subset and h : W → R
m+1
be a smooth function. Define f : W → R by f(x) = (x,h(x)). Then f is
an injective immersion with f = (e ,h ), where e is the i-th standard vector
i i i i
m
in R . For all p ∈ M, the tangent space Tpf is the linear hull of the linearly
independent vectors f (p). If m = 2, N f is the line determined by the vector
i p
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