Equivalence of two ways of defining a Riemannian metric
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Definition 1 A Riemannian metric on a smooth manifold is smooth family of inner products on the tangent spaces of M. So g is Riemannian metric if it assigns to each point $p in M$ a positive definite symmetric bilinear form on $T_p M$,
$$ g_p: T_p M times T_p M rightarrow mathbb{R} $$
with smoothness referring to the requirement that the function
$$ p mapsto g_p(X_p, Y_p)$$
must be smooth for any locally defined vector fields X,Y in M.
Definition 2 A Riemannian metric on a smooth manifold X is a smooth section of $S^2 T^* X subset otimes^2 T^*X$ such that $left.fright|_X in S^2 T^*_xX$ is a positive definite quadratic form on $T_xX$.
Why are these two definitions equivalent?
geometry manifolds riemannian-geometry
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add a comment |
$begingroup$
Definition 1 A Riemannian metric on a smooth manifold is smooth family of inner products on the tangent spaces of M. So g is Riemannian metric if it assigns to each point $p in M$ a positive definite symmetric bilinear form on $T_p M$,
$$ g_p: T_p M times T_p M rightarrow mathbb{R} $$
with smoothness referring to the requirement that the function
$$ p mapsto g_p(X_p, Y_p)$$
must be smooth for any locally defined vector fields X,Y in M.
Definition 2 A Riemannian metric on a smooth manifold X is a smooth section of $S^2 T^* X subset otimes^2 T^*X$ such that $left.fright|_X in S^2 T^*_xX$ is a positive definite quadratic form on $T_xX$.
Why are these two definitions equivalent?
geometry manifolds riemannian-geometry
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Just write down both definitions in local coordinates; then you will see that the two notions are equivalent. Keep in mind that to verify 1 it suffices to check it for (local) coordinate vector fields.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 4:34
add a comment |
$begingroup$
Definition 1 A Riemannian metric on a smooth manifold is smooth family of inner products on the tangent spaces of M. So g is Riemannian metric if it assigns to each point $p in M$ a positive definite symmetric bilinear form on $T_p M$,
$$ g_p: T_p M times T_p M rightarrow mathbb{R} $$
with smoothness referring to the requirement that the function
$$ p mapsto g_p(X_p, Y_p)$$
must be smooth for any locally defined vector fields X,Y in M.
Definition 2 A Riemannian metric on a smooth manifold X is a smooth section of $S^2 T^* X subset otimes^2 T^*X$ such that $left.fright|_X in S^2 T^*_xX$ is a positive definite quadratic form on $T_xX$.
Why are these two definitions equivalent?
geometry manifolds riemannian-geometry
$endgroup$
Definition 1 A Riemannian metric on a smooth manifold is smooth family of inner products on the tangent spaces of M. So g is Riemannian metric if it assigns to each point $p in M$ a positive definite symmetric bilinear form on $T_p M$,
$$ g_p: T_p M times T_p M rightarrow mathbb{R} $$
with smoothness referring to the requirement that the function
$$ p mapsto g_p(X_p, Y_p)$$
must be smooth for any locally defined vector fields X,Y in M.
Definition 2 A Riemannian metric on a smooth manifold X is a smooth section of $S^2 T^* X subset otimes^2 T^*X$ such that $left.fright|_X in S^2 T^*_xX$ is a positive definite quadratic form on $T_xX$.
Why are these two definitions equivalent?
geometry manifolds riemannian-geometry
geometry manifolds riemannian-geometry
asked Dec 20 '18 at 10:17
gengen
4982521
4982521
$begingroup$
Just write down both definitions in local coordinates; then you will see that the two notions are equivalent. Keep in mind that to verify 1 it suffices to check it for (local) coordinate vector fields.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 4:34
add a comment |
$begingroup$
Just write down both definitions in local coordinates; then you will see that the two notions are equivalent. Keep in mind that to verify 1 it suffices to check it for (local) coordinate vector fields.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 4:34
$begingroup$
Just write down both definitions in local coordinates; then you will see that the two notions are equivalent. Keep in mind that to verify 1 it suffices to check it for (local) coordinate vector fields.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 4:34
$begingroup$
Just write down both definitions in local coordinates; then you will see that the two notions are equivalent. Keep in mind that to verify 1 it suffices to check it for (local) coordinate vector fields.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 4:34
add a comment |
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$begingroup$
Just write down both definitions in local coordinates; then you will see that the two notions are equivalent. Keep in mind that to verify 1 it suffices to check it for (local) coordinate vector fields.
$endgroup$
– Moishe Cohen
Dec 27 '18 at 4:34