Riemannian metric
Deﬁnition (Riemannian metric)
Let M be an m-dimensional smooth manifold and let
g ∈ Λ
1
(T
?
M) ⊗ Λ
1
(T
?
M) be such that for an open set U ∈ M
and a system of coordinates x = (x
1
, . . . , x
m
) in U, we have
g
ij
(p) = g(∂
i
, ∂
j
) = h∂
i
, ∂
j
i,
for all p ∈ U and i, j = 1, . . . , m. The tangent vectors
∂
1
, . . . , ∂
m
∈ T
p
M are assumed to form an orthonormal frame.
The Riemannian metric g
p
(v, w) at p ∈ M is the inner product
between the tangent vectors v, w ∈ T
p
M.