In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold which is calculated from the metric tensor on a larger manifold into which the submanifold is embedded. It may be calculated using the following formula (written using Einstein summation convention):
g a b = ∂ a X μ ∂ b X ν g μ ν Here a , b describe the indices of coordinates ξ a of the submanifold while the functions X μ ( ξ a ) encode the embedding into the higher-dimensional manifold whose tangent indices are denoted μ , ν .
Let
Π : C → R 3 , τ ↦ { x 1 = ( a + b cos ( n ⋅ τ ) ) cos ( m ⋅ τ ) x 2 = ( a + b cos ( n ⋅ τ ) ) sin ( m ⋅ τ ) x 3 = b sin ( n ⋅ τ ) . be a map from the domain of the curve C with parameter τ into the euclidean manifold R 3 . Here a , b , m , n ∈ R are constants.
Then there is a metric given on R 3 as
g = ∑ μ , ν g μ ν d x μ ⊗ d x ν with g μ ν = ( 1 0 0 0 1 0 0 0 1 ) .
and we compute
g τ τ = ∑ μ , ν ∂ x μ ∂ τ ∂ x ν ∂ τ g μ ν ⏟ δ μ ν = ∑ μ ( ∂ x μ ∂ τ ) 2 = m 2 a 2 + 2 m 2 a b cos ( n ⋅ τ ) + m 2 b 2 cos 2 ( n ⋅ τ ) + b 2 n 2 Therefore g C = ( m 2 a 2 + 2 m 2 a b cos ( n ⋅ τ ) + m 2 b 2 cos 2 ( n ⋅ τ ) + b 2 n 2 ) d τ ⊗ d τ
