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LSpice
LSpice
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The Sard-Smale result certainly guarantees that this will be true for a generic 𝑔, but will it hold for any 𝑔?

If $g$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence of a Morse function $f$ so that the pair $(f,g)$ is Morse-Smale.

And yes, there's also a proof with less heavy machinery, see for instance Theorem 6.6 in the book "Lecture Notes on Morse Homology""Lectures on Morse Homology" by Banyaga and Hurtubise.

The Sard-Smale result certainly guarantees that this will be true for a generic 𝑔, but will it hold for any 𝑔?

If $g$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence of a Morse function $f$ so that the pair $(f,g)$ is Morse-Smale.

And yes, there's also a proof with less heavy machinery, see for instance Theorem 6.6 in the book "Lecture Notes on Morse Homology" by Banyaga and Hurtubise.

The Sard-Smale result certainly guarantees that this will be true for a generic 𝑔, but will it hold for any 𝑔?

If $g$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence of a Morse function $f$ so that the pair $(f,g)$ is Morse-Smale.

And yes, there's also a proof with less heavy machinery, see for instance Theorem 6.6 in the book "Lectures on Morse Homology" by Banyaga and Hurtubise.

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Alessio Pellegrini
Alessio Pellegrini
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The Sard-Smale result certainly guarantees that this will be true for a generic 𝑔, but will it hold for any 𝑔?

If $g$ is fixed, you can certainly use the Sard-Smale theorem to prove the existence of a Morse function $f$ so that the pair $(f,g)$ is Morse-Smale.

And yes, there's also a proof with less heavy machinery, see for instance Theorem 6.6 in the book "Lecture Notes on Morse Homology" by Banyaga and Hurtubise.