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Pete L. Clark
Pete L. Clark
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[Edited for for punctuation, grammar and clarity -- PLC]

Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a Euclidean space of sufficiently large dimension, and we thus get a Riemannian metric on $M$ by restricting the Euclidean metric on the ambient space.

Can all Riemannian metrics on $M$ be constructed in the second way above?

[Edited for for punctuation, grammar and clarity -- PLC]

[Edited for for punctuation, grammar and clarity -- PLC]

Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a Euclidean space of sufficiently large dimension, and we thus get a Riemannian metric on $M$ by restricting the Euclidean metric on the ambient space.

Can all Riemannian metrics on $M$ be constructed in the second way above?

Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a Euclidean space of sufficiently large dimension, and we thus get a Riemannian metric on $M$ by restricting the Euclidean metric on the ambient space.

Can all Riemannian metrics on $M$ be constructed in the second way above?

[Edited for for punctuation, grammar and clarity -- PLC]

deleted 24 characters in body; edited title
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Pete L. Clark
Pete L. Clark
  • 65.4k
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does Are all riemannian metricRiemannian metrics induced by euclideanEuclidean metrics? [Nash Embedding Theorem]

we all know that we[Edited for for punctuation, grammar and clarity -- PLC]

Let $M$ be a smooth manifold. We can get a riemannianRiemannian metric on $M$ by at least two method ,one ismethods: first by the philosophy of the partitionpartitions of the unit,while the other isunity and second by the whitneyWhitney embedding theorem ,we: we can embed the riemannian manifold to an euclidean$M$ into a Euclidean space of sufficiently large dimension,and we and we thus get a riemannianRiemannian metric on $M$ by the restriction ofrestricting the canonical euclidean metri toEuclidean metric on the tangent vector bundleambient space. Then what i want to ask is that whether

Can all riemannianRiemannian metrics canon $M$ be constructed in the second way of above?

does all riemannian metric induced by euclidean metrics?

we all know that we can get a riemannian metric by at least two method ,one is by the philosophy of the partition of the unit,while the other is by the whitney embedding theorem ,we can embed the riemannian manifold to an euclidean space of sufficiently large dimension,and we thus get a riemannian metric by the restriction of the canonical euclidean metri to the tangent vector bundle. Then what i want to ask is that whether all riemannian metrics can be constructed in the second way of above?

Are all Riemannian metrics induced by Euclidean metrics? [Nash Embedding Theorem]

[Edited for for punctuation, grammar and clarity -- PLC]

Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a Euclidean space of sufficiently large dimension, and we thus get a Riemannian metric on $M$ by restricting the Euclidean metric on the ambient space.

Can all Riemannian metrics on $M$ be constructed in the second way above?

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HKSHLZW
HKSHLZW
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does all riemannian metric induced by euclidean metrics?

we all know that we can get a riemannian metric by at least two method ,one is by the philosophy of the partition of the unit,while the other is by the whitney embedding theorem ,we can embed the riemannian manifold to an euclidean space of sufficiently large dimension,and we thus get a riemannian metric by the restriction of the canonical euclidean metri to the tangent vector bundle. Then what i want to ask is that whether all riemannian metrics can be constructed in the second way of above?