1. Recall that a real vector space V is called Euclidean if it is equipped with a scalar
product, that is, a symmetric bilinear form such that
(v, v) > 0 for any v 6= 0.
The standard scalar product on Rn is
(x, y) =
n
X
xi yi .
i=1
Prove that the volume of the parallelepiped in Rn defined by v1 , . . . , vn ∈ Rn is
p
det(A), where Aij = (vi , vj ).
2. A vector bundle ξ = (E, M, π) is called Euclidean if its every fiber Ex = π −1 (x)
is a Euclidean space such that for any smooth sections s1 , s2 of ξ the function M 3 x 7→
(s1 (x), s2 (x)) is smooth.
Show that a Euclidean structure on ξ defines a section of the bundle ξ ∗ ⊗ ξ ∗ . Using
partitions of unity conclude that any vector bundle has a Euclidean structure (see Theorem 3.3.7).
3. A manifold M is called Riemannian if its tangent bundle τ (M ) is equipped with a
Euclidean structure. By the previous exercise any manifold can be made Riemannian.
Let M m be an oriented Riemannian manifold. If v1 , . . . , vm is an oriented basis in τp (M ),
set
p
ω(v1 , . . . , vm ) = det(A), where Aij = (vi , vj ).
Show that this way we get a volume form on M .
4. Let M m ⊂ Rn . Then τp (M ) ⊂ τp (Rn ) = Rn , so τp (M ) has a canonical scalar product
for any p ∈ M . Thus M is a Riemannian manifold.
Assume M = f (U ), where f : U → Rn is a regular map, U ⊂ Rm . Prove that for the
corresponding volume form ω on M we have
f ∗ (ω) =
n
X
p
∂fk
∂fk
det(A(x)) dx1 ∧ . . . ∧ dxm , where A(x)i,j =
(x)
(x).
∂x
∂x
i
j
k=1
5. Compute the expression for the volume form on S 2 ⊂ R3 in terms of the coordinates
given by
(i) stereographic projection;
(ii) spherical coordinates.