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Manifolds
Adrian Down
December 07, 2006
1 Introduction
1.1 Motivation
Today, we continue our exploration of higher-dimensional geometry. We will need many more definitions to build up geometric structure on manifolds. The higher-dimensional theory will not make reference to the space in which the manifold in embedded (this space is R^3 , for example, in the case that the manifold is a surface). Some of the important formulas presented earlier in the course will reappear, with the summation indices now from 1 to N instead of from 1 to 3. Once we have established the requisite material, we will go through a famous calculation of the orbit of Mercury. The calculation uses general relativity and was first performed by Einstein.
1.2 Klein bottle
The Klein bottle is a special type of closed surface that does not separate space into a region “interior” to the surface and a region “exterior” to the surface. The Klein bottle is difficult to visualize, but can be realized more simply by considering a planar model. Consider a rectangular region. Let the top and bottom correspond so that if a curve moves off the top or bottom of the region, the curve continues at the same horizontal position on the other side of the rectangle (this corresponds to “gluing” the top and bottom sides together when constructing a surface). If the remaining two sides of the figure have the same correspondence, the resulting figure is a torus. Instead, let the remaining two sides have reflection
symmetry relative to each other, so that moving off near the top left corner, for example, the curve reappears in the bottom right corner of the figure. We can compute the Euler characteristic of the klein bottle by connecting the two corners with curves to form an “X” over the rectangular region. The region is then divided into four faces. The segments of the “X” divide each other in two, and thus contribute four edges to χ. The boundaries of the rectangular region contribute only two more edges, since edges that are glued together represent only one edge on the actual figure. By the way in which the figure is connected, all of the corners of the rectangle correspond to the same vertex, so there are only two independent vertices. The Euler characteristic of the klein bottle is thus,
χ = 4 − 6 + 2 = 0
1.3 Review
Our goal in defining manifolds will be to make no reference to extrinsic quantities.
- A set M is called an n-dimensional manifold if every point p ∈ M can be covered with a coordinate chart (U, φ).
- F(M ) was defined to be the set of all smooth functions,
F = { f : M → R | f is smooth }
- Tangent vectors are defined to be operators that operate on smooth functions. A function Xp is a tangent vector if it satisfies the following properties, - Xp(f + g) = Xpf + Xpg for all f, g ∈ F(M ) - Xp(rf ) = rXpf - Xp(f g) = f (p)Xpg + g(p)Xp(f )
Tangent vectors are thought of as a form of differentiation. We can form a basis of the tangent space of the form, ( ∂ ∂xi
p
f =
∂ui
f ◦ φ−^1
(φ(p))
For any function f ∈ F(M ), the Lie bracket can be computed,
[X, Y ]f = x^1 x^2
∂x^1
x^2
∂f ∂x^2
− x^2
∂f ∂x^2
x^1 x^2
∂f ∂x^1
= x^1
x^2
) 2 ∂^2 f ∂x^1 x^2
− x^1 x^2
∂f ∂x^1
− x^1
x^2
) 2 ∂^2 f ∂x^1 x^2 = −x^1 x^2
∂f ∂x^2
Since this holds for any f ∈ F(M ),
[X, Y ] = −x^1 x^2
∂x^2
Theorem. If X, Y ∈ X(M ), then [X, Y ] ∈ X(M )
Note. This theorem says that for vector fields X and Y , [X, Y ] is also a vector field. However, for f ∈ F(M ), X(Y f ) and Y (Xf ) are not vector fields independently.
2.3 Differentials
Definition (Differential). Let M and N be manifolds, and let Φ : M → N. The differential of Φ at a point p ∈ M is the function,
(Φ∗)p : TpM → TΦ(p)N
given by, ( (Φ∗)p Xp
f = Xp (f ◦ Φ) for all f ∈ F(N )
Note. The important content of this definition is that Φ∗ “pushes forward” tangent vectors on M to tangent vectors on N.
Example. Let α : T → M be a curve. Then,
Tα ∈ TαM
The tangent vector is related to the time derivative,
α∗
d dt
= Tα
2.4 Connections
Connections are the the key definition of the theory thus far. They provide a method for differentiating one vector field into another.
Definition (Connection). A connection is a mapping ∆ : X(M ) × X(M ) → X(M ) that satisfies,
- ∆X is linear in the upper argument,
∆X (Y + Z) = ∆X Y + ∆X (Z)
- For r ∈ R,
∆X (rY ) = r∆X Y
- ∆X is also linear in the lower argument,
∆X+Y Z = ∆X Z + ∆Y Z
- For a function f ∈ F(M ), define,
∆f X Y = f ∆X Y
Koszol showed that ∆X satisfies,
∆X (f Y ) = (Xf )Y + f ∆X Y
Note. ∆X Y is called the covariant derivative of Y in the direction of X.
Definition (Christoffel symbols). Let (U, φ) be a chart on M , and TP M be the tangent space at p ∈ M ,
TpM = span
∂xi
∣ i^ = 1,... , n
The Christoffel symbols are defined as,
∂xi
∂xj
∑^ n
k=
Γkij
∂xk
Note. The important point of this definition is that the Christoffel symbols Γkij are the expressions of the basis (^) ∂x∂i in a particular basis. This is the connection between the abstract geometric theory and calculations performed in a particular chart.
4 Metric coefficients
4.1 Riemann metric
Until now, we have not encountered the analog of the metric coefficients gij. These coefficients depended on the inner product. We assume that such a metric is provided on the manifold.
Definition (Riemann metric). A Riemann metric g on a surface M is a mapping,
g : X(M ) × X(M ) → F(M )
that satisfies the following properties,
- Linearity
g(X + Y, Z) = g(X, Y ) + g(Y, Z) g(rX, Y ) = rg(X, Y )
- Symmetry
g(X, Y ) = g(Y, X)
- Positive semi-definiteness,
g(X, X) ≥ 0 g(X, X) = 0 ⇔ X = 0
Note. The metric can be written in local coordinates. Consider a chart (U, φ) and a tangent space,
TpM = span
∂xi
∣ i^ = 1,... , n
The metric coefficients are defined,
gij ≡ g
∂xi^
∂xj
4.2 Christoffel symbols
We would like to construct the Christoffel symbols from the metric coeffi- cients. To do so, we will need the following definitions.
Definition (Metrical). ∆ is called metrical if,
Xg(Y, Z) = g (∆X Y, Z) + g (Y, ∆X Z)
Definition (Torsion free). ∆ is torsion free if,
∆X Y − ∆Y X = [X, Y ]
Theorem. Let M be a surface with a Riemann metric g. Then there exists a unique, metrical, torsion free connection ∆
Note. The importance of this theorem is it allows us to express Γkij in terms of the metric. Recall the definition of Γkij in terms of local coordinates,
∂xi
∂xj
∑^ n
k=
Γkij
∂xk
We can then define the Christoffel symbols using the metric,
Γkij ≡
∑^ n
l=
gkl
∂gil ∂uj^
∂gjl ∂ui^
∂gij ∂ul
4.3 Riemann curvature tensor
We can now use the previous definitions to define the Riemann curvature tensor, which appeared unmotivated in the proof that the Gaussian curvature K is intrinsic.
Definition (Riemann curvature tensor). The Riemann curvature tensor is a function,
R : X(M ) × X(M ) × X(M ) → X(M )
which satisfies,
R(X, Y )Z = ∆X ∆Y Z − ∆Y ∆X Z − ∆[X,Y ]Z
