218
IX – Multivariate Differential and Integral Calculus
§ 4. Differential Manifolds
This § gives a very summary treatment of the simplest aspects of the theory
of differential manifolds. Even just considering its most basic notions, it has
several other aspects. There are many excellent presentations of the subject
and far more complete than ours.
46
11 – What is a Manifold ?
(i) The sphere in R
3 . To understand this problem, consider the unit sphere
X having equation x
2 + y
2 + z
2 = 1 in R
3 . What would be a reasonable way
to define differential functions on an open subset of X ?
A first condition they need to satisfy is to be continuous and defined by
properties of a local nature. Then consider a point (a, b, c) ∈ X and a function f defined and continuous in the neighbourhood of this point. Suppose
that c > 0. The upper hemisphere H + : z > 0 of X is an open subset of X
and also the graph of a C
∞ function
z =
1 − x
2
− y
2
1/2 ,
defined on the open subset x
2 + y
2 < 1 of R
2 . Setting
ϕ(x, y, z) = (x, y) ,
define a homeomorphism from H + onto an open subset of R
2 transforming
f into a function of (x, y), defined and continuous in the neighbourhood of
the point ϕ(a, b, c) ∈ R
2 . It is then natural to say that f is of class C
r in the
neighbourhood of (a, b, c) if, as a function of (x, y), it is of class C
r in the
classical sense. We adopt the same convention if c < 0, i.e. if we are in the
lower hemisphere H − of X, the graph of the function
z = −
1 − x
2
− y
2
1/2 .
If c = 0, perhaps b < 0 ; then replace H + by the hemisphere y < 0, which is
the graph of the equation
y = −
1 − x
2
− z
2
1/2 .
Formula ϕ(x, y, z) = (x, z) again defines a homeomorphism from this hemisphere onto an open subset of the plane and differentiable function in the
46 Marcel Berger and Bernard Gostiaux, G´ eom´ etrie diff´ erentielle (A. Colin, 1972),
Paul Malliavin, G´ eom´ etrie diff´ erentielle intrins` eque (Hermann, 1972), Pham
Mau Quan, Introduction ` a la g´ eom´ etrie des vari´ et´ es diff´ erentiables (Dunod,
1969), Frank W. Warner, Foundations of Differential Manifolds and Lie Groups
(Scott, Foresman, 1971), Michael Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Inc, 5 vol.), Shlomo Sternberg, Lectures
on Differential Geometry (Prentice-Hall, 1964), Serge Lang, Fundamentals of
Differential Geometry (Springer, 1999).
IX – Multivariate Differential and Integral Calculus
§ 4. Differential Manifolds
This § gives a very summary treatment of the simplest aspects of the theory
of differential manifolds. Even just considering its most basic notions, it has
several other aspects. There are many excellent presentations of the subject
and far more complete than ours.
46
11 – What is a Manifold ?
(i) The sphere in R
3 . To understand this problem, consider the unit sphere
X having equation x
2 + y
2 + z
2 = 1 in R
3 . What would be a reasonable way
to define differential functions on an open subset of X ?
A first condition they need to satisfy is to be continuous and defined by
properties of a local nature. Then consider a point (a, b, c) ∈ X and a function f defined and continuous in the neighbourhood of this point. Suppose
that c > 0. The upper hemisphere H + : z > 0 of X is an open subset of X
and also the graph of a C
∞ function
z =
1 − x
2
− y
2
1/2 ,
defined on the open subset x
2 + y
2 < 1 of R
2 . Setting
ϕ(x, y, z) = (x, y) ,
define a homeomorphism from H + onto an open subset of R
2 transforming
f into a function of (x, y), defined and continuous in the neighbourhood of
the point ϕ(a, b, c) ∈ R
2 . It is then natural to say that f is of class C
r in the
neighbourhood of (a, b, c) if, as a function of (x, y), it is of class C
r in the
classical sense. We adopt the same convention if c < 0, i.e. if we are in the
lower hemisphere H − of X, the graph of the function
z = −
1 − x
2
− y
2
1/2 .
If c = 0, perhaps b < 0 ; then replace H + by the hemisphere y < 0, which is
the graph of the equation
y = −
1 − x
2
− z
2
1/2 .
Formula ϕ(x, y, z) = (x, z) again defines a homeomorphism from this hemisphere onto an open subset of the plane and differentiable function in the
46 Marcel Berger and Bernard Gostiaux, G´ eom´ etrie diff´ erentielle (A. Colin, 1972),
Paul Malliavin, G´ eom´ etrie diff´ erentielle intrins` eque (Hermann, 1972), Pham
Mau Quan, Introduction ` a la g´ eom´ etrie des vari´ et´ es diff´ erentiables (Dunod,
1969), Frank W. Warner, Foundations of Differential Manifolds and Lie Groups
(Scott, Foresman, 1971), Michael Spivak, A Comprehensive Introduction to Differential Geometry (Publish or Perish, Inc, 5 vol.), Shlomo Sternberg, Lectures
on Differential Geometry (Prentice-Hall, 1964), Serge Lang, Fundamentals of
Differential Geometry (Springer, 1999).