Differential Geometry of Manifolds Riemannian Metrics Lengths and Volumes, guaranteed and verified 100% Pass

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Riemannian Metrics – Lengths and Volumes

Now that we have a metric on a Riemannian manifold, we can investigate
how to measure lengths of curves and “volumes” of regions on manifolds (note:
“volume” means length for a curve, surface area for a surface, volume for a 3-
dimensional region, and 𝑛-dimensional volume for an 𝑛-dimensional region).

We know from second year calculus that the length of a curve in ℝ3 (using
the standard metric on ℝ3 ) is given by:
𝑏
𝑙 (𝛾 ) = ∫ ‖𝛾 ′ (𝑡)‖ 𝑑𝑡
𝑎
where the curve, 𝛾, is given by:

𝛾(𝑡) = < 𝑥 (𝑡), 𝑦(𝑡), 𝑧(𝑡) >; 𝑎 ≤ 𝑡 ≤ 𝑏.

We are going to generalize this definition to apply to curves on
𝑛-dimensional manifolds with a Riemannian metric, 𝑔. Suppose we can
parametrize an 𝑛-dimensional manifold, 𝑀, by:
⃗Φ
⃗⃗ : 𝑈 ⊆ ℝ𝑛 → 𝑀 ⊆ ℝ𝑘 .

We can think of any curve, 𝛾, that lays on 𝑀 as the image under Φ ⃗⃗⃗ of a
curve, 𝛼, in 𝑈 ⊆ ℝ𝑛 . ⃗⃗⃗ (𝑢1 (𝑡), … , 𝑢𝑛 (𝑡))
𝛾(𝑡) = Φ



𝛼(𝑡) = (𝑢1 (𝑡), … , 𝑢𝑛 (𝑡)) 𝑀
⃗⃗⃗
Φ

𝑈



So 𝛾 (𝑡) = ⃗Φ
⃗⃗ (𝑢1 (𝑡), … , 𝑢𝑛 (𝑡)), where 𝛼 (𝑡) = (𝑢1 (𝑡), … , 𝑢𝑛 (𝑡)).

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By the Chain Rule:
1 𝑛
⃗⃗ 𝑢1 𝑑𝑢 + ⋯ + ⃗Φ
𝛾 ′ (𝑡) = ⃗Φ ⃗⃗ 𝑢𝑛 𝑑𝑢
𝑑𝑡 𝑑𝑡
So we can write:

‖𝛾 ′ (𝑡)‖2 = 𝛾 ′ (𝑡) ⋅ 𝛾 ′ (𝑡)
1 𝑛 1 𝑛
⃗⃗⃗ 𝑢1 𝑑𝑢 + ⋯ + ⃗Φ
= (Φ ⃗⃗ 𝑢𝑛 𝑑𝑢 ) ⋅ (Φ
⃗⃗⃗ 𝑢1 𝑑𝑢 + ⋯ + ⃗Φ
⃗⃗ 𝑢𝑛 𝑑𝑢 )
𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡
𝑛
𝑑𝑢𝑖 𝑑𝑢 𝑗
= ∑ 𝑔𝑖𝑗 = < 𝛾 ′ (𝑡), 𝛾 ′ (𝑡) >
𝑑𝑡 𝑑𝑡
𝑖,𝑗=1

where < , > is the inner product from the Riemannian metric 𝑔.




Notice that here we have used the metric induced by the parametrization ⃗Φ
⃗⃗ of
𝑀. However, we might have a Riemannian metric that doesn’t come from the
⃗⃗⃗ , but the following definition still holds.
parametrization Φ




Def. We define the length of a curve, 𝛾, on a Riemannian manifold, (𝑀, 𝑔), by:


𝑏 𝑏 𝑛
𝑑𝑢𝑖 𝑑𝑢 𝑗
𝑙(𝛾 ) = ∫ √< 𝛾 ′ (𝑡), 𝛾 ′ (𝑡) > 𝑑𝑡 = ∫ √ ∑ 𝑔𝑖𝑗 𝑑𝑡.
𝑎 𝑎 𝑑𝑡 𝑑𝑡
𝑖,𝑗=1