Notes on Riemannian Geometry

The first variational formula

Lemma 6.3: symmetry lemma
Lemma 6.4: every V is a variation field of some γ
Proposition 6.5: the first variational formula
Theorem 6.6: every minimizing curve is geodesic
Corollary 6.7: γ critical point of L iff γ geodesic

How to get ε from the compactness property in Lemma 6.4: first consider a covering of $γ ([a, b])$ with uniformly normal neighborhood of each $γ (t)$ and extract a finite covering. Let $V_{max} = max_{t \in [a, b]} V (t)$ and let $ε = inf \frac{δ_{n}}{V_{max}}$ where $δ_{n}$ are the numbers associated to the selected uniformly normal neighborhoods. If we consider the geodesic $σ$ starting at $γ (t)$ and with inital speed $\frac{V (t)}{V_{max}}$ then it is slower than the corresponding unit speed (which is well defined in a geodesic ball centered in $γ (t)$ and of radius $ε . V_{max}$ ) so is defined at least between times $\pm ε . V_{max}$ . Hence $Γ (s, t) = \exp_{γ (t)} (sV (t)) = σ (s . V_{max})$ is well defined.

The Gauß' Lemma and its consequences

Theorem 6.8: Gauß' Lemma
Corollary 6.9: $grad r = \frac{\partial}{\partial r}$
Corollary 6.10: in geodesic ball, the radial geodesic is the unique minimizing curve.
Corollary 6.11: $r (x) = d (p, x)$
Theorem 6.12: every riemannian geodesic is locally minimizing
Theorem 6: every minimizing curve is geodesic (alternative proof)

Hopf & Rinow

We consider the following statements:

M is compact
M complete as metric space
M geodesically complete i.e. for all $p \in M$ , $\exp_{p}$ is defined on the whole tangent space $T_{p} M$ .
Any two points $p, q$ in M can be joined by a minimizing segment.

We also introduce the weaker forms C' and D' where we replace in C and D "for all p" by "there exists a p". We have the following graph of implications ( $C \Rightarrow C'$ and $D \Rightarrow D'$ are trivial):

1 A 2 B 1->2 3 C 1->3 6.16 2->3 4 D 2->4 6.15 3->4 5 C' 3->5 6 D' 4->6 5->2 6.14 5->6

The open disc is an example of a non-complete manifold with property D, the converse direction in corollary 6.15 is false. Similarly, the converse of corollary 6.16 is false, because $ℝ^{n}$ is an example of a noncompact complete riemannian manifold.

Theorem 6.13 shows that B and C are equivalent in three steps:

$B \Rightarrow C$
$C' \Rightarrow D'$ . Note that the same $p$ is used, so actually we can generalize to $C \Rightarrow D$
$C' \Rightarrow B$ . We use D' to build the sequence $γ_{i}$ (so that $d_{i}$ is a Cauchy sequence). However we don't have $D \Rightarrow A$ as stated in corollary 6.15, because we also need C to ensure that $\exp_{p} V$ is well-defined.

$A \Rightarrow B$ holds because any compact metric space is complete. This should allow to get corollaries 6.15 and 6.16 from the graph.

The Curvature Tensor - Flat manifolds

Proposition 7.1: curvature endomorphism is tensorial
Lemma 7.2: curvature endomorphism and tensor are intrisic properties
Theorem 7.3: manifold flat iff curvature tensor vanishes

In proof of theorem 7.3, "So it suffices to show that $\nabla_{\partial_{k + 1}} (\nabla_{\partial_{i}} E_{j}) = 0$ " is because we already know $\nabla_{\partial_{i}} E_{j} = 0$ on $M_{k}$ , so if $\nabla_{\partial_{i}} E_{j}$ is parallel along the $k + 1$ direction then $\nabla_{\partial_{i}} E_{j} = 0$ on the whole $M_{k + 1}$ .

Symmetries of Curvature Tensor

Proposition 7.4: symmetries of curvature tensor

The rotation angle theorem

Theorem 9.1: the rotation angle theorem

The Gauß-Bonnet Formula

Lemma 9.2: $γ$ polygon in $M$ , $Rot (γ) = 2 π$
Theorem 9.3: the Gauß-Bonnet formula.
Corollary 9.4: using $K = 0$ , $κ_{n} = 0$ , $ε_{i} = π - α_{i}$ we get $\sum_{i} α_{i} = π$
Corollary 9.5: using $K = 0$ , $κ_{N} = \frac{1}{R}$ and $ε_{i} = 0$ , we get $L = 2 π R$ .
Corollary 9.6: $K = 0$ and $ε_{i} = 0$
Another example given during the lecture for the sphere : let $Area (α)$ be the area of a digon of interior angle $α$ . It is proportional to $α$ and by evaluating (for example) at $α = \frac{π}{2}$ we get $Area (α) = 2 α$ . If $D$ is the area of the triangle given by the intersection of three such digons, and A, B, C denotes the area of the remaining pieces, we have: $A + B + C + D = \frac{4 π}{2}$ and $A + D = 2 α$ , $B + D = 2 β$ and $C + D = 2 γ$ . Finally we get $A + B + C + D = 2 π = 2 (α + β + γ + D)$ so $D = α + β + γ - π$ . This is also what we get using the Gauß-Bonnet formula, with $K = 0$ , $κ_{N} = \frac{1}{1}$ and $ε_{1} = π - α$ , $ε_{2} = π - β$ and $ε_{3} = π - γ$ .

End of the proof of theorem theorem 9.3, as given in the lecture. First we define:

$\int_{Ω} f ⅆ A = \int_{φ (Ω)} (f \circ φ^{-1}) \sqrt{\det g} ⅆ x_{1} ⅆ x_{2}$
$K = \frac{Rm (\partial_{1}, \partial_{2}, \partial_{2}, \partial_{1})}{\det g}$
$dω (X, Y) = X (ω (Y)) - Y (ω (X)) - ω ([X, Y])$
$\int_{γ} P ⅆ x + Q ⅆ y = \int_{a}^{b} P \frac{ⅆ γ_{1}}{ⅆ t} + Q \frac{ⅆ γ_{2}}{ⅆ t} ⅆ t$

We construct an orthonormal basis $(\begin{matrix} E_{1} \\ E_{2} \end{matrix}) = A (\begin{matrix} \partial_{1} \\ \partial_{2} \end{matrix})$ using Gram-Schmidt. From definition 2 and 3, we check $\det g {(\det A)}^{2} = \det g^{E} = 1$ and $K = ⅆ ω (E_{1}, E_{2}) = (\det A) ⅆ ω (\partial_{1}, \partial_{2})$ and consequently, by definition 1:

$\int_{Ω} K ⅆ A = \int_{φ (Ω)} ⅆ ω (\partial_{1}, \partial_{2}) ⅆ x_{1} ⅆ x_{2}$

For coordinate frames, $[\partial_{1}, \partial_{2}] = 0$ so $ⅆ ω (\partial_{1}, \partial_{2}) = \frac{\partial ω_{2}}{\partial x_{1}} - \frac{\partial ω_{1}}{\partial x_{2}}$ where $ω_{i} = ω (\partial_{i})$ . Using Green's Theorem and definition 4 we get the expected result:

$\int_{Ω} K ⅆ A = \int_{a}^{b} ω_{1} \frac{ⅆ γ_{1}}{ⅆ t} + ω_{2} \frac{ⅆ γ_{2}}{ⅆ t} ⅆ t = \int_{a}^{b} ω (\frac{ⅆ γ_{1}}{ⅆ t} \partial_{1} + \frac{ⅆ γ_{2}}{ⅆ t} \partial_{2}) ⅆ t = \int_{a}^{b} ω (\dot{γ} (t)) ⅆ t$

The Gauß-Bonnet Theorem

Theorem 9.7: The Gauß-Bonnet theorem
Corollary: $χ (M)$ does not depend on triangulation. $χ (M) = 0$ for the torus, $χ (M) = 2$ for the sphere. The only possible values are $2 - 2 g$ (stated but not proved during the lecture).
If there is a nonzero vector field, then $χ (M) = 0$ . (hence there is no nonzero vector on the sphere) proof: If $X$ is a nonzero vector field on the whole manifold, we can define $E_{1} = \frac{X}{∥ X ∥}$ and $E_{2}$ such that $(E_{1}, E_{2})$ is orthonormal. We have a 1-form $ω$ such that $\nabla_{Y} E_{1} = - ω (Y) E_{2}$ and $\nabla_{Y} E_{2} = ω (Y) E_{1}$ for any vector field $Y$ , as in the proof of the Gauß-Bonnet Formula. $\int_{M} K ⅆ A = \int_{M} ⅆ ω = \sum_{i} \int_{Δ_{i}} ω = \sum_{i} \int_{\partial Δ_{i}} ω = 0$ ( $ω$ is defined on the whole manifold and as in the proof of the Gauß-Bonnet Theorem, the integrals on the edges cancel out).
If $K \leq 0$ , there is no simple digon. proof: suppose the contrary, let $α_{1}, α_{2} > 0$ be the interior angles. Then $ε_{i} = π - α < π$ , $κ_{n} = 0$ and $\int_{M} K ⅆ A = 2 π - ε_{1} - ε_{2} > 0$ .
If $K > 0$ , any two geodesics intersect. proof: otherwise we can cut the manifold into three disjoint connected pieces $Ω_{1}, Ω_{2}, Ω_{3}$ using this geodesics. We have $χ (Ω_{i}) = \frac{1}{2 π} \int_{ω_{i}} K ⅆ A > 0$ and since it is an integer, $χ (Ω_{i}) \geq 1$ . Hence $χ (M) = \sum_{i = 1}^{3} χ (Ω_{i}) \geq 3$ , which is a contradiction (according to the possible values of 2).

Only theorem 9.7 is from Lee's book, the others have only been given during the lecture. The other propositions of chapter 9 have not been seen during the lecture.

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