First Basic Forms, Riemannian Metrics
📂Geometry First Basic Forms, Riemannian Metrics Buildup Riemannian metric is a concept that comes from the process of calculating the length of curves on a surface, and the process is as follows.
Let’s say α ( t ) \boldsymbol{\alpha}(t) α ( t ) regular curve moving on a simple surface x : U → R 3 \mathbf{x} : U \to \mathbb{R}^{3} x : U → R 3 ( u 1 , u 2 ) (u_{1}, u_{2}) ( u 1 , u 2 ) U U U α \boldsymbol{\alpha} α
α ( t ) = x ( u 1 ( t ) , u 2 ( t ) )
\boldsymbol{\alpha}(t) = \mathbf{x}(u_{1}(t), u_{2}(t))
α ( t ) = x ( u 1 ( t ) , u 2 ( t ))
At this point, the length of α \boldsymbol{\alpha} α a ≤ t ≤ b a \le t \le b a ≤ t ≤ b
∫ a b ∣ d α d t ∣ d t
\int_{a}^{b} \left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| dt
∫ a b d t d α d t
If we resolve the integrand function, it goes as follows.
∣ d α d t ∣ = ⟨ d α d t , d α d t ⟩ = ⟨ d x ( u 1 , u 2 ) d t , d x ( u 1 , u 2 ) d t ⟩
\begin{align*}
\left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| =&\ \sqrt{\left\langle \dfrac{d \boldsymbol{\alpha}}{d t} , \dfrac{d \boldsymbol{\alpha}}{d t} \right\rangle}
\\ =&\ \sqrt{\left\langle \dfrac{d \mathbf{x}(u_{1}, u_{2})}{d t} , \dfrac{d \mathbf{x}(u_{1}, u_{2})}{d t} \right\rangle}
\end{align*}
d t d α = = ⟨ d t d α , d t d α ⟩ ⟨ d t d x ( u 1 , u 2 ) , d t d x ( u 1 , u 2 ) ⟩
By the chain rule ,
∣ d α d t ∣ = ⟨ ∂ x ∂ u 1 d u 1 d t + ∂ x ∂ u 2 d u 2 d t , ∂ x ∂ u 1 d u 1 d t + ∂ x ∂ u 2 d u 2 d t ⟩ = ⟨ x 1 d u 1 d t + x 2 d u 2 d t , x 1 d u 1 d t + x 2 d u 2 d t ⟩
\begin{align*}
\left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| =&\ \sqrt{\left\langle \dfrac{\partial \mathbf{x}}{\partial u_{1}}\dfrac{d u_{1}}{dt} + \dfrac{\partial \mathbf{x}}{\partial u_{2}}\dfrac{d u_{2}}{dt}, \dfrac{\partial \mathbf{x}}{\partial u_{1}}\dfrac{d u_{1}}{dt} + \dfrac{\partial \mathbf{x}}{\partial u_{2}}\dfrac{d u_{2}}{dt} \right\rangle}
\\ =&\ \sqrt{\left\langle \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt}, \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt} \right\rangle}
\end{align*}
d t d α = = ⟨ ∂ u 1 ∂ x d t d u 1 + ∂ u 2 ∂ x d t d u 2 , ∂ u 1 ∂ x d t d u 1 + ∂ u 2 ∂ x d t d u 2 ⟩ ⟨ x 1 d t d u 1 + x 2 d t d u 2 , x 1 d t d u 1 + x 2 d t d u 2 ⟩
At this point, x 1 : = ∂ x ∂ u 1 , x 2 : = ∂ x ∂ u 2 \mathbf{x}_{1} := \dfrac{\partial \mathbf{x}}{\partial u_{1}}, \mathbf{x}_{2} := \dfrac{\partial \mathbf{x}}{\partial u_{2}} x 1 := ∂ u 1 ∂ x , x 2 := ∂ u 2 ∂ x
∣ d α d t ∣ = ⟨ x 1 d u 1 d t + x 2 d u 2 d t , x 1 d u 1 d t + x 2 d u 2 d t ⟩ = ( d u 1 d t ) 2 ⟨ x 1 , x 1 ⟩ + d u 1 d t d u 2 d t ⟨ x 1 , x 2 ⟩ + d u 2 d t d u 1 d t ⟨ x 2 , x 1 ⟩ + ( d u 2 d t ) 2 ⟨ x 2 , x 2 ⟩
\begin{align*}
& \left| \dfrac{d \boldsymbol{\alpha}}{d t} \right|
\\ =&\ \sqrt{\left\langle \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt}, \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt} \right\rangle}
\\ =&\ \sqrt{\left( \dfrac{d u_{1}}{dt} \right)^{2} \left\langle \mathbf{x}_{1}, \mathbf{x}_{1} \right\rangle + \dfrac{d u_{1}}{dt}\dfrac{d u_{2}}{dt} \left\langle \mathbf{x}_{1}, \mathbf{x}_{2} \right\rangle + \dfrac{d u_{2}}{dt}\dfrac{d u_{1}}{dt} \left\langle \mathbf{x}_{2}, \mathbf{x}_{1} \right\rangle + \left( \dfrac{d u_{2}}{dt} \right)^{2} \left\langle \mathbf{x}_{2}, \mathbf{x}_{2} \right\rangle}
\end{align*}
= = d t d α ⟨ x 1 d t d u 1 + x 2 d t d u 2 , x 1 d t d u 1 + x 2 d t d u 2 ⟩ ( d t d u 1 ) 2 ⟨ x 1 , x 1 ⟩ + d t d u 1 d t d u 2 ⟨ x 1 , x 2 ⟩ + d t d u 2 d t d u 1 ⟨ x 2 , x 1 ⟩ + ( d t d u 2 ) 2 ⟨ x 2 , x 2 ⟩
Here, if we denote the dot product above by g i j = ⟨ x i , x j ⟩ g_{ij} = \left\langle \mathbf{x}_{i}, \mathbf{x}_{j} \right\rangle g ij = ⟨ x i , x j ⟩ ∑ \sum ∑
∣ d α d t ∣ = ∑ i = 1 2 ∑ j = 1 2 g i j d u i d t d u j d t = g i j d u i d t d u j d t
\begin{align*}
\left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| =&\ \sqrt{ \sum \limits_{i=1}^{2}\sum \limits_{j=1}^{2} g_{ij} \dfrac{d u_{i}}{dt}\dfrac{d u_{j}}{dt}}
\\ =&\ \sqrt{ g_{ij} \dfrac{d u_{i}}{dt}\dfrac{d u_{j}}{dt}}
\end{align*}
d t d α = = i = 1 ∑ 2 j = 1 ∑ 2 g ij d t d u i d t d u j g ij d t d u i d t d u j
In the second equality, the summation sign is omitted using Einstein notation .
Definition g i j = ⟨ x i , x j ⟩ g_{ij} = \left\langle \mathbf{x}_{i}, \mathbf{x}_{j} \right\rangle g ij = ⟨ x i , x j ⟩ Riemannian metric , or the coefficient of the first fundamental form .
Let M M M surface at R 3 \mathbb{R}^{3} R 3 p ∈ M p \in M p ∈ M X , Y \mathbf{X}, \mathbf{Y} X , Y tangent vector at p p p eigenmap x : U → R 3 \mathbf{x} : U \to \mathbb{R}^{3} x : U → R 3 M M M expressed as follows .
X = X 1 x 1 + X 2 x 2 and Y = Y 1 x 1 + Y 2 x 2
\mathbf{X} = X^{1}\mathbf{x}_{1} + X^{2}\mathbf{x}_{2} \quad \text{and} \quad \mathbf{Y} = Y^{1}\mathbf{x}_{1} + Y^{2}\mathbf{x}_{2}
X = X 1 x 1 + X 2 x 2 and Y = Y 1 x 1 + Y 2 x 2
The following bilinear form I I I Riemannian metric of the surface x \mathbf{x} x the first fundamental form .
I : T p M × T p M → R
I : T_{p}M \times T_{p}M \to \mathbb{R}
I : T p M × T p M → R
I ( X , Y ) = ∑ i = 1 2 ∑ j = 1 2 g i j X i Y j = g i j X i Y j = [ X 1 X 2 ] [ g 11 g 12 g 21 g 22 ] [ Y 1 Y 2 ]
I (\mathbf{X}, \mathbf{Y}) = \sum \limits_{i=1}^{2} \sum \limits_{j=1}^{2} g_{ij}X^{i}Y^{j} = g_{ij}X^{i}Y^{j}
= \begin{bmatrix} X^{1} & X^{2}\end{bmatrix}
\begin{bmatrix}
g_{11} & g_{12} \\ g_{21} & g_{22}
\end{bmatrix}
\begin{bmatrix} Y^{1} \\ Y^{2}\end{bmatrix}
I ( X , Y ) = i = 1 ∑ 2 j = 1 ∑ 2 g ij X i Y j = g ij X i Y j = [ X 1 X 2 ] [ g 11 g 21 g 12 g 22 ] [ Y 1 Y 2 ]
The determinant of the matrix of coefficients [ g i j ] \left[ g_{ij} \right] [ g ij ] g g g
g : = det ( [ g i j ] ) = ∣ g 11 g 12 g 21 g 22 ∣ = g 11 g 22 − g 12 g 21
g := \det (\left[ g_{ij} \right]) = \begin{vmatrix} g_{11} & g_{12} \\ g_{21} & g_{22}\end{vmatrix} = g_{11}g_{22} - g_{12}g_{21}
g := det ( [ g ij ] ) = g 11 g 21 g 12 g 22 = g 11 g 22 − g 12 g 21
The ( k , l ) (k,l) ( k , l ) inverse matrix of the matrix [ g i j ] \left[ g_{ij} \right] [ g ij ] g k l g^{kl} g k l
( g 11 g 12 g 21 g 22 ) − 1 = 1 det [ g i j ] ( g 22 − g 21 g 12 g 22 ) = 1 g ( g 22 − g 21 g 12 g 22 ) = ( g 22 g − g 21 g − g 12 g g 11 g ) = ( g 11 g 12 g 21 g 22 )
\begin{align*}
\begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} ^{-1}
=&\ \dfrac{1}{\det \left[ g_{ij} \right]} \begin{pmatrix} g_{22} & - g_{21} \\ g_{12} & g_{22} \end{pmatrix}
= \dfrac{1}{g} \begin{pmatrix} g_{22} & - g_{21} \\ g_{12} & g_{22} \end{pmatrix}
\\[1em] =&\ \begin{pmatrix}\dfrac{g_{22}}{g} & - \dfrac{g_{21}}{g} \\[1em] -\dfrac{g_{12}}{g} & \dfrac{g_{11}}{g} \end{pmatrix}
\\[1em] =&\ \begin{pmatrix} g^{11} & g^{12} \\[1em] g^{21} & g^{22} \end{pmatrix}
\end{align*}
( g 11 g 21 g 12 g 22 ) − 1 = = = det [ g ij ] 1 ( g 22 g 12 − g 21 g 22 ) = g 1 ( g 22 g 12 − g 21 g 22 ) g g 22 − g g 12 − g g 21 g g 11 g 11 g 21 g 12 g 22
Explanation These days, the term “first fundamental form” is hardly used and mostly, the term “Riemannian metric” is used. The name “metric” is used because, as seen in the buildup, it is used for measuring the length of curves on the surface.
Notations like E = g 11 E = g_{11} E = g 11 F = g 21 = g 12 F=g_{21}=g_{12} F = g 21 = g 12 G = g 22 G=g_{22} G = g 22
The reason the concept of Riemannian metric, which did not appear in curve theory, arises is because the basis of the tangent space { x 1 , x 2 } \left\{ \mathbf{x}_{1}, \mathbf{x}_{2} \right\} { x 1 , x 2 } orthonormal basis . If it were orthonormal, g i j = δ i j g_{ij} = \delta_{ij} g ij = δ ij δ \delta δ Kronecker delta . Using the Riemannian metric and Einstein notation, the length of the curve α \boldsymbol{\alpha} α
L ( α ) = length of α = ∫ a b g i j d u i d t d u j d t d t = ∫ a b g i j α i ′ α j ′ d t = ∫ a b E ( d u 1 d t ) 2 + 2 F d u 1 d t d u 2 d t + G ( d u 2 d t ) 2 d t
\begin{align*}
L (\boldsymbol{\alpha}) =&\ \text{length of } \boldsymbol{\alpha}
\\ =&\ \int_{a}^{b} \sqrt{ g_{ij} \dfrac{d u_{i}}{dt}\dfrac{d u_{j}}{dt}} dt
\\ =&\ \int_{a}^{b} \sqrt{ g_{ij} \alpha_{i}^{\prime} \alpha_{j}^{\prime} } dt
\\ =&\ \int_{a}^{b} \sqrt{ E\left( \dfrac{d u_{1}}{dt} \right)^{2} + 2F\dfrac{d u_{1}}{dt}\dfrac{d u_{2}}{dt} + G\left( \dfrac{d u_{2}}{dt} \right)^{2}} dt
\end{align*}
L ( α ) = = = = length of α ∫ a b g ij d t d u i d t d u j d t ∫ a b g ij α i ′ α j ′ d t ∫ a b E ( d t d u 1 ) 2 + 2 F d t d u 1 d t d u 2 + G ( d t d u 2 ) 2 d t
The area of the surface is also defined by the integral of the Riemannian metric.
For a certain region R R R x \mathbf{x} x Q = x − 1 ( R ) Q = \mathbf{x}^{-1}(R) Q = x − 1 ( R ) Q ⊂ U ⊂ R 2 Q \subset U \subset \R^{2} Q ⊂ U ⊂ R 2 R R R area of R = ∬ Q g d u 1 d u 2 = ∬ Q ∣ x 1 × x 2 ∣ d u 1 d u 2 = ∬ Q E G − F 2 d u 1 d u 2
\text{area of } R = \iint _{Q} \sqrt{g} du_{1}du_{2} = \iint _{Q} \left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right| du_{1}du_{2} = \iint _{Q} \sqrt{EG-F^{2}} du_{1}du_{2}
area of R = ∬ Q g d u 1 d u 2 = ∬ Q ∣ x 1 × x 2 ∣ d u 1 d u 2 = ∬ Q EG − F 2 d u 1 d u 2
Properties For a simple surface x : U → R 3 \mathbf{x} : U \to \mathbb{R}^{3} x : U → R 3
(a) g = ∣ x 1 × x 2 ∣ 2 g = \left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right|^{2} g = ∣ x 1 × x 2 ∣ 2
(b) g 11 = g 22 g and g 12 = g 21 = − g 12 g and g 22 = g 11 g g^{11} = \dfrac{g_{22}}{g} \quad \text{and} \quad g^{12} = g^{21} = -\dfrac{g_{12}}{g} \quad \text{and} \quad g^{22} = \dfrac{g_{11}}{g} g 11 = g g 22 and g 12 = g 21 = − g g 12 and g 22 = g g 11
(c) ∀ i , j \forall i,j ∀ i , j ∑ k = 1 2 g i k g k j = δ i j \sum \limits_{k=1}^{2} g_{ik}g^{kj} = {\delta_{i}}^{j} k = 1 ∑ 2 g ik g kj = δ i j
Here, δ \delta δ Kronecker delta .
Proof (a) Due to the properties of the cross product and the definition of Riemannian metric, the following holds.
∣ x 1 × x 2 ∣ 2 = ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 sin 2 θ = ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 ( 1 − cos 2 θ ) = ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 ( 1 − x 1 ⋅ x 2 ∣ x 1 ∣ ∣ x 2 ∣ ) = ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 − ( x 1 ⋅ x 2 ) 2 = g 11 g 22 − g 12 g 21 = det ( [ g i j ] ) = g
\begin{align*}
\left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right|^{2} =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2} \sin ^{2} \theta
\\ =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2}\left(1- \cos ^{2} \theta \right)
\\ =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2}\left(1- \dfrac{\mathbf{x}_{1} \cdot \mathbf{x}_{2}}{\left| \mathbf{x}_{1} \right| \left| \mathbf{x}_{2} \right| } \right)
\\ =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2} - \left( \mathbf{x}_{1} \cdot \mathbf{x}_{2} \right)^{2}
\\ =&\ g_{11}g_{22} - g_{12}g_{21}
\\ =&\ \det( [g_{ij}] )
\\ =&\ g
\end{align*}
∣ x 1 × x 2 ∣ 2 = = = = = = = ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 sin 2 θ ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 ( 1 − cos 2 θ ) ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 ( 1 − ∣ x 1 ∣ ∣ x 2 ∣ x 1 ⋅ x 2 ) ∣ x 1 ∣ 2 ∣ x 2 ∣ 2 − ( x 1 ⋅ x 2 ) 2 g 11 g 22 − g 12 g 21 det ([ g ij ]) g
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(b) It’s according to the definition.
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(c) Since [ g k l ] [g^{kl}] [ g k l ] [ g i j ] [g_{ij}] [ g ij ]
( 1 0 0 1 ) = ( g 11 g 12 g 21 g 22 ) ( g 11 g 12 g 21 g 22 ) = ( g 11 g 11 + g 12 g 21 g 11 g 12 + g 12 g 22 g 21 g 11 + g 22 g 21 g 21 g 12 + g 22 g 22 )
\begin{align*}
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} =&\ \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} \begin{pmatrix} g^{11} & g^{12} \\ g^{21} & g^{22} \end{pmatrix}
\\[1em] =&\ \begin{pmatrix} g_{11}g^{11}+g_{12}g^{21} & g_{11}g^{12} + g_{12}g^{22} \\[1em] g_{21}g^{11} + g_{22}g^{21} & g_{21}g^{12} + g_{22}g^{22} \end{pmatrix}
\end{align*}
( 1 0 0 1 ) = = ( g 11 g 21 g 12 g 22 ) ( g 11 g 21 g 12 g 22 ) g 11 g 11 + g 12 g 21 g 21 g 11 + g 22 g 21 g 11 g 12 + g 12 g 22 g 21 g 12 + g 22 g 22
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See Also 