Geodesic Curvature is Intrinsic
📂Geometry Geodesic Curvature is Intrinsic Theorem The geodesic curvature κ g \kappa_{g} κ g surface is intrinsic .
Description In other words, κ g \kappa_{g} κ g unit normal n \mathbf{n} n extrinsic formula , as κ N = T ′ = α ′ ′ = κ n n + κ g S \kappa \mathbf{N} = \mathbf{T}^{\prime} = \alpha^{\prime \prime} = \kappa_{n}\mathbf{n}+ \kappa_{g}\mathbf{S} κ N = T ′ = α ′′ = κ n n + κ g S
κ g = ⟨ T ′ , S ⟩ = ⟨ T ′ , n × T ⟩ = [ T ′ , n , T ] = [ n , T , T ′ ] = ⟨ n , T × T ′ ⟩ = ⟨ n , T × κ N ⟩ = κ ⟨ n , T × N ⟩ = κ ⟨ n , B ⟩ = κ cos θ
\begin{align*}
\kappa_{g} =&\ \left\langle \mathbf{T}^{\prime}, \mathbf{S} \right\rangle
\\ =&\ \left\langle \mathbf{T}^{\prime}, \mathbf{n} \times \mathbf{T} \right\rangle
\\ =&\ \left[ \mathbf{T}^{\prime}, \mathbf{n}, \mathbf{T} \right]
\\ =&\ \left[ \mathbf{n}, \mathbf{T}, \mathbf{T}^{\prime} \right]
\\ =&\ \left\langle \mathbf{n}, \mathbf{T} \times \mathbf{T}^{\prime} \right\rangle
\\ =&\ \left\langle \mathbf{n}, \mathbf{T} \times \kappa \mathbf{N} \right\rangle
\\ =&\ \kappa \left\langle \mathbf{n}, \mathbf{T} \times \mathbf{N} \right\rangle
\\ =&\ \kappa \left\langle \mathbf{n}, \mathbf{B} \right\rangle
\\ =&\ \kappa \cos \theta
\end{align*}
κ g = = = = = = = = = ⟨ T ′ , S ⟩ ⟨ T ′ , n × T ⟩ [ T ′ , n , T ] [ n , T , T ′ ] ⟨ n , T × T ′ ⟩ ⟨ n , T × κ N ⟩ κ ⟨ n , T × N ⟩ κ ⟨ n , B ⟩ κ cos θ
Here, [ ⋅ , ⋅ , ⋅ ] \left[ \cdot, \cdot, \cdot \right] [ ⋅ , ⋅ , ⋅ ] scalar triple product . B \mathbf{B} B binormal . θ \theta θ n \mathbf{n} n B \mathbf{B} B
Proof g i j g_{ij} g ij Riemann metric L i j = ⟨ x i j , n ⟩ L_{ij} = \left\langle \mathbf{x}_{ij}, \mathbf{n} \right\rangle L ij = ⟨ x ij , n ⟩ second fundamental form Γ i j k = ∑ l = 1 2 ⟨ x i j , x l ⟩ g l k = ⟨ x i j , x l ⟩ g l k \Gamma_{ij}^{k} = \sum \limits_{l=1}^{2} \left\langle \mathbf{x}_{ij}, \mathbf{x}_{l} \right\rangle g^{lk} = \left\langle \mathbf{x}_{ij}, \mathbf{x}_{l} \right\rangle g^{lk} Γ ij k = l = 1 ∑ 2 ⟨ x ij , x l ⟩ g l k = ⟨ x ij , x l ⟩ g l k Christoffel symbols Let x : U → R 3 \mathbf{x} : U \to \mathbb{R}^{3} x : U → R 3 coordinate chart mapping , and the coordinates of U U U ( u 1 , u 2 ) (u_{1}, u_{2}) ( u 1 , u 2 ) α ( s ) = x ( u 1 ( s ) , u 2 ( s ) ) \alpha (s) = \mathbf{x}\left( u_{1}(s), u_{2}(s) \right) α ( s ) = x ( u 1 ( s ) , u 2 ( s ) ) α ′ ′ \alpha^{\prime \prime} α ′′ expressed as follows .
α ′ ′ ( s ) = κ n ( s ) n ( s ) + κ g ( s ) S ( s )
\alpha^{\prime \prime}(s) = \kappa_{n}(s)\mathbf{n}(s) + \kappa_{g}(s)\mathbf{S}(s)
α ′′ ( s ) = κ n ( s ) n ( s ) + κ g ( s ) S ( s )
n \mathbf{n} n unit normal , S = n × T \mathbf{S} = \mathbf{n} \times \mathbf{T} S = n × T x i \mathbf{x}_{i} x i x i j \mathbf{x}_{ij} x ij x \mathbf{x} x
x i : = ∂ x ∂ u i and x i j : = ∂ 2 x ∂ u j ∂ u i
\mathbf{x}_{i} := \dfrac{\partial \mathbf{x}}{\partial u_{i}} \quad \text{and} \quad
\mathbf{x}_{ij} := \dfrac{\partial^{2} \mathbf{x}}{\partial u_{j} \partial u_{i}}
x i := ∂ u i ∂ x and x ij := ∂ u j ∂ u i ∂ 2 x
Part 1.
First, let’s represent the scalar triple product of unit normals and the bases of the tangent plane as follows.
ϵ i j = ⟨ n , x i × x j ⟩ = [ n , x i , x j ]
\epsilon _{ij} = \langle \mathbf{n}, \mathbf{x}_{i} \times \mathbf{x}_{j} \rangle = \left[ \mathbf{n}, \mathbf{x}_{i}, \mathbf{x}_{j} \right]
ϵ ij = ⟨ n , x i × x j ⟩ = [ n , x i , x j ]
Then, since x i × x i = 0 \mathbf{x}_{i} \times \mathbf{x}_{i} = \mathbf{0} x i × x i = 0
ϵ 11 = ϵ 22 = 0
\epsilon_{11} = \epsilon_{22} = 0
ϵ 11 = ϵ 22 = 0
Since n \mathbf{n} n x 1 × x 2 \mathbf{x}_{1} \times \mathbf{x}_{2} x 1 × x 2 definition ,
⟨ n , x 1 × x 2 ⟩ = ∣ n ∣ ∣ x 1 × x 2 ∣
\langle \mathbf{n}, \mathbf{x}_{1} \times \mathbf{x}_{2} \rangle = \left| \mathbf{n} \right| \left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right|
⟨ n , x 1 × x 2 ⟩ = ∣ n ∣ ∣ x 1 × x 2 ∣
Here, since n \mathbf{n} n g = ∣ x 1 × x 2 ∣ 2 g = \left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right|^{2} g = ∣ x 1 × x 2 ∣ 2
ϵ 12 = ⟨ n , x 1 × x 2 ⟩ = − ⟨ n , x 2 × x 1 ⟩ = − ϵ 21 = g
\epsilon_{12} = \langle \mathbf{n}, \mathbf{x}_{1} \times \mathbf{x}_{2} \rangle = -\langle \mathbf{n}, \mathbf{x}_{2} \times \mathbf{x}_{1} \rangle = -\epsilon_{21} = \sqrt{g}
ϵ 12 = ⟨ n , x 1 × x 2 ⟩ = − ⟨ n , x 2 × x 1 ⟩ = − ϵ 21 = g
Part 2.
Since S = n × T \mathbf{S} = \mathbf{n} \times \mathbf{T} S = n × T S \mathbf{S} S
κ g = ⟨ k g S , S ⟩ = ⟨ k g S , n × T ⟩ = [ κ g S , n , T ]
\kappa _{g} = \left\langle k_{g} \mathbf{S}, \mathbf{S} \right\rangle = \left\langle k_{g} \mathbf{S}, \mathbf{n} \times \mathbf{T} \right\rangle = \left[ \kappa_{g}\mathbf{S}, \mathbf{n}, \mathbf{T} \right]
κ g = ⟨ k g S , S ⟩ = ⟨ k g S , n × T ⟩ = [ κ g S , n , T ]
Formula
κ g S = ∑ i = 1 2 [ u k ′ ′ + ∑ i , j = 1 2 Γ i j k u i ′ u j ′ ] x k
\kappa_{g}\mathbf{S} = \sum \limits_{i=1}^{2} \left[ u_{k}^{\prime \prime} + \sum \limits_{i,j=1}^{2} \Gamma_{ij}^{k}u_{i}^{\prime}u_{j}^{\prime} \right] \mathbf {x}_{k}
κ g S = i = 1 ∑ 2 [ u k ′′ + i , j = 1 ∑ 2 Γ ij k u i ′ u j ′ ] x k
The tangent vector is T = x l u l ′ \mathbf{T} = \mathbf{x}_{l}u_{l}^{\prime} T = x l u l ′ κ g \kappa_{g} κ g
κ g = ⟨ k g S , n × T ⟩ = [ κ g S , n , T ] = ⟨ ∑ k = 1 2 ( u k ′ ′ + ∑ i = 1 2 ∑ j = 1 2 Γ i j k u i ′ u j ′ ) x k , n × T ⟩ = ∑ k = 1 2 ( u k ′ ′ + ∑ i = 1 2 ∑ j = 1 2 Γ i j k u i ′ u j ′ ) ⟨ x k , n × T ⟩ = ∑ k = 1 2 ( u k ′ ′ + ∑ i = 1 2 ∑ j = 1 2 Γ i j k u i ′ u j ′ ) [ x k , n , T ] = ∑ k = 1 2 ( u k ′ ′ + ∑ i = 1 2 ∑ j = 1 2 Γ i j k u i ′ u j ′ ) [ n , T , x k ] = ∑ k = 1 2 ( u k ′ ′ + ∑ i = 1 2 ∑ j = 1 2 Γ i j k u i ′ u j ′ ) [ n , x l u l ′ , x k ] = ∑ k = 1 2 ( u k ′ ′ + ∑ i = 1 2 ∑ j = 1 2 Γ i j k u i ′ u j ′ ) u l ′ [ n , x l , x k ] = ∑ k = 1 2 ( u k ′ ′ + ∑ i = 1 2 ∑ j = 1 2 Γ i j k u i ′ u j ′ ) u l ′ ϵ l k
\begin{align*}
\kappa_{g} =&\ \left\langle k_{g} \mathbf{S}, \mathbf{n} \times \mathbf{T} \right\rangle = \left[ \kappa_{g}\mathbf{S}, \mathbf{n}, \mathbf{T} \right]
\\ =&\ \left\langle \sum \limits_{k=1}^{2} \left( u^{\prime \prime}_{k} + \sum\limits_{i=1}^{2} \sum\limits_{j=1}^{2} \Gamma_{ij}^{k} u_{i}^{\prime} u_{j}^{\prime} \right)\mathbf{x}_{k} , \mathbf{n} \times \mathbf{T} \right\rangle
\\ =&\ \sum \limits_{k=1}^{2} \left( u^{\prime \prime}_{k} + \sum\limits_{i=1}^{2} \sum\limits_{j=1}^{2} \Gamma_{ij}^{k} u_{i}^{\prime} u_{j}^{\prime} \right) \left\langle \mathbf{x}_{k} , \mathbf{n} \times \mathbf{T} \right\rangle
\\ =&\ \sum \limits_{k=1}^{2} \left( u^{\prime \prime}_{k} + \sum\limits_{i=1}^{2} \sum\limits_{j=1}^{2} \Gamma_{ij}^{k} u_{i}^{\prime} u_{j}^{\prime} \right) \left[\mathbf{x}_{k}, \mathbf{n}, \mathbf{T} \right]
\\ =&\ \sum \limits_{k=1}^{2} \left( u^{\prime \prime}_{k} + \sum\limits_{i=1}^{2} \sum\limits_{j=1}^{2} \Gamma_{ij}^{k} u_{i}^{\prime} u_{j}^{\prime} \right) \left[\mathbf{n}, \mathbf{T}, \mathbf{x}_{k} \right]
\\ =&\ \sum \limits_{k=1}^{2} \left( u^{\prime \prime}_{k} + \sum\limits_{i=1}^{2} \sum\limits_{j=1}^{2} \Gamma_{ij}^{k} u_{i}^{\prime} u_{j}^{\prime} \right) \left[\mathbf{n}, \mathbf{x}_{l}u_{l}^{\prime}, \mathbf{x}_{k} \right]
\\ =&\ \sum \limits_{k=1}^{2} \left( u^{\prime \prime}_{k} + \sum\limits_{i=1}^{2} \sum\limits_{j=1}^{2} \Gamma_{ij}^{k} u_{i}^{\prime} u_{j}^{\prime} \right) u_{l}^{\prime} \left[\mathbf{n}, \mathbf{x}_{l}, \mathbf{x}_{k} \right]
\\ =&\ \sum \limits_{k=1}^{2} \left( u^{\prime \prime}_{k} + \sum\limits_{i=1}^{2} \sum\limits_{j=1}^{2} \Gamma_{ij}^{k} u_{i}^{\prime} u_{j}^{\prime} \right) u_{l}^{\prime} \epsilon_{lk}
\end{align*}
κ g = = = = = = = = ⟨ k g S , n × T ⟩ = [ κ g S , n , T ] ⟨ k = 1 ∑ 2 ( u k ′′ + i = 1 ∑ 2 j = 1 ∑ 2 Γ ij k u i ′ u j ′ ) x k , n × T ⟩ k = 1 ∑ 2 ( u k ′′ + i = 1 ∑ 2 j = 1 ∑ 2 Γ ij k u i ′ u j ′ ) ⟨ x k , n × T ⟩ k = 1 ∑ 2 ( u k ′′ + i = 1 ∑ 2 j = 1 ∑ 2 Γ ij k u i ′ u j ′ ) [ x k , n , T ] k = 1 ∑ 2 ( u k ′′ + i = 1 ∑ 2 j = 1 ∑ 2 Γ ij k u i ′ u j ′ ) [ n , T , x k ] k = 1 ∑ 2 ( u k ′′ + i = 1 ∑ 2 j = 1 ∑ 2 Γ ij k u i ′ u j ′ ) [ n , x l u l ′ , x k ] k = 1 ∑ 2 ( u k ′′ + i = 1 ∑ 2 j = 1 ∑ 2 Γ ij k u i ′ u j ′ ) u l ′ [ n , x l , x k ] k = 1 ∑ 2 ( u k ′′ + i = 1 ∑ 2 j = 1 ∑ 2 Γ ij k u i ′ u j ′ ) u l ′ ϵ l k
Here, since Γ i j k \Gamma_{ij}^{k} Γ ij k ϵ l k \epsilon_{lk} ϵ l k κ g \kappa_{g} κ g
