Equivalence of two ways of defining a Riemannian metric












1












$begingroup$


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?










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$endgroup$












  • $begingroup$
    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
















1












$begingroup$


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?










share|cite|improve this question









$endgroup$












  • $begingroup$
    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














1












1








1


1



$begingroup$


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?










share|cite|improve this question









$endgroup$




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






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asked Dec 20 '18 at 10:17









gengen

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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


















  • $begingroup$
    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










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