Mixing
deformation of Hodge star operator and harmonic forms
$begingroup$
Suppose $(M,g)$ is a compact Riemannian manifold, and $*_g$ is the Hodge star operator defined on the de Rham algebra $Omega^*(M)$ with respect to the metric $g$. Let $phi:Mto M$ be a diffeomorphism. Then, can we find another metric $h$ (which should be related to $phi$) so that
$$
(phi^{-1})^* circ *_gcirc phi^*
$$
agree with the new Hodge star operator $*_h$?
EDIT: I guess this should be the Hodge star operator for the pullback metric. On the other hand, we know $phi$ induces a map $$
phi^*:H^*(M)to H^*(M)$$ on de Rham cohomology groups, and meanwhile we also know for any metric $h$ there is the so-called Hodge isomorphism $mathcal H^*_h(M) to H^*(M)$ where $mathcal H^*_h(M)$ denotes the $h$-harmonic forms. Now another interesting question is that can we find a metric $h$ so that we can induce a map
$$ phi^*: mathcal H_g(M) to mathcal H_h(M)$$?
differential-geometry differential-topology riemannian-geometry hodge-theory de-rham-cohomology
asked Jan 6 at 2:23
HangHang
482315
$endgroup$
add a comment |
$begingroup$
Suppose $(M,g)$ is a compact Riemannian manifold, and $*_g$ is the Hodge star operator defined on the de Rham algebra $Omega^*(M)$ with respect to the metric $g$. Let $phi:Mto M$ be a diffeomorphism. Then, can we find another metric $h$ (which should be related to $phi$) so that
$$
(phi^{-1})^* circ *_gcirc phi^*
$$
agree with the new Hodge star operator $*_h$?
EDIT: I guess this should be the Hodge star operator for the pullback metric. On the other hand, we know $phi$ induces a map $$
phi^*:H^*(M)to H^*(M)$$ on de Rham cohomology groups, and meanwhile we also know for any metric $h$ there is the so-called Hodge isomorphism $mathcal H^*_h(M) to H^*(M)$ where $mathcal H^*_h(M)$ denotes the $h$-harmonic forms. Now another interesting question is that can we find a metric $h$ so that we can induce a map
$$ phi^*: mathcal H_g(M) to mathcal H_h(M)$$?
differential-geometry differential-topology riemannian-geometry hodge-theory de-rham-cohomology
asked Jan 6 at 2:23
HangHang
482315
$endgroup$
add a comment |
$begingroup$
Suppose $(M,g)$ is a compact Riemannian manifold, and $*_g$ is the Hodge star operator defined on the de Rham algebra $Omega^*(M)$ with respect to the metric $g$. Let $phi:Mto M$ be a diffeomorphism. Then, can we find another metric $h$ (which should be related to $phi$) so that
$$
(phi^{-1})^* circ *_gcirc phi^*
$$
agree with the new Hodge star operator $*_h$?
EDIT: I guess this should be the Hodge star operator for the pullback metric. On the other hand, we know $phi$ induces a map $$
phi^*:H^*(M)to H^*(M)$$ on de Rham cohomology groups, and meanwhile we also know for any metric $h$ there is the so-called Hodge isomorphism $mathcal H^*_h(M) to H^*(M)$ where $mathcal H^*_h(M)$ denotes the $h$-harmonic forms. Now another interesting question is that can we find a metric $h$ so that we can induce a map
$$ phi^*: mathcal H_g(M) to mathcal H_h(M)$$?
differential-geometry differential-topology riemannian-geometry hodge-theory de-rham-cohomology
asked Jan 6 at 2:23
HangHang
482315
$endgroup$
Suppose $(M,g)$ is a compact Riemannian manifold, and $*_g$ is the Hodge star operator defined on the de Rham algebra $Omega^*(M)$ with respect to the metric $g$. Let $phi:Mto M$ be a diffeomorphism. Then, can we find another metric $h$ (which should be related to $phi$) so that
$$
(phi^{-1})^* circ *_gcirc phi^*
$$
agree with the new Hodge star operator $*_h$?
EDIT: I guess this should be the Hodge star operator for the pullback metric. On the other hand, we know $phi$ induces a map $$
phi^*:H^*(M)to H^*(M)$$ on de Rham cohomology groups, and meanwhile we also know for any metric $h$ there is the so-called Hodge isomorphism $mathcal H^*_h(M) to H^*(M)$ where $mathcal H^*_h(M)$ denotes the $h$-harmonic forms. Now another interesting question is that can we find a metric $h$ so that we can induce a map
$$ phi^*: mathcal H_g(M) to mathcal H_h(M)$$?
differential-geometry differential-topology riemannian-geometry hodge-theory de-rham-cohomology
differential-geometry differential-topology riemannian-geometry hodge-theory de-rham-cohomology
asked Jan 6 at 2:23
HangHang
482315
asked Jan 6 at 2:23
HangHang
482315
edited Jan 9 at 16:20
Hang
asked Jan 6 at 2:23
HangHang
482315
asked Jan 6 at 2:23
HangHang
482315
asked Jan 6 at 2:23
HangHang
482315
482315
add a comment |
add a comment |
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