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my issue seems pretty trivial, however, I cannot find a way.

I create a manifold, with 2 frames/charts, and create transition_map between them. Create a metric. I create then a submanifold, create diff_map to the first manifold, and make an embedding. Then I try to get the induced_metric.

It works only if the diff_map is referenced to the first created frame/chart at top level.

In my exemple : top manifold with cartesian and cylindrical charts sub manifold is a cylinder. I cannot base it on the cylindrical top chart. My purpose would be to have a top manifold series of charts at top and natural embeddings of lower dimension manifolds. Any idea?

My code : E = Manifold(3, 'E', structure='Riemannian') cartesian.<x,y,z>=E.chart() cartesian_frame=cartesian.frame()

cylindrical.<r,ph,z> = E.chart(r'r:(0,+oo) ph:(0,2pi):\phi z') cylindrical_frame=cylindrical.frame()
cyl2cart = cylindrical.transition_map(cartesian , [rcos(ph),r*sin(ph),z]) cart2cyl = cyl2cart.set_inverse(sqrt(x^2+y^2), atan2(y,x),z)

ecc=E.coord_changes() for cc in ecc : display(ecc[cc].display())

g=E.metric() g[cartesian_frame, 0,0],g[cartesian_frame,1,1],g[cartesian_frame,2,2]=1,1,1 display(g.display(cartesian)) display(g.display(cylindrical))

C2 = Manifold(2, 'C2',ambient=E, structure='Riemannian') Cyl2.<ph,z>=C2.chart(r' ph:\phi z') Cyl2_frame=Cyl2.frame()

if True : ### True shows issue E.set_default_frame(cylindrical_frame) E.set_default_chart(cylindrical) Cyl2tocylindrical=C2.diff_map(E,{(Cyl2,cylindrical):[1,ph,z]}) cylindricaltoCyl2=E.diff_map(C2,{(cylindrical,Cyl2):[ph,z]}) C2.set_embedding(Cyl2tocylindrical)

else : Cyl2tocartesian=C2.diff_map(E,{(Cyl2,cartesian):[cos(ph),sin(ph),z]}) cartesiantoCyl2=E.diff_map(C2,{(cartesian,Cyl2):[arctan2(y,x),z]}) C2.set_embedding(Cyl2tocartesian)

C2g=C2.induced_metric() ### Never an error message here display(C2g.display()) ### exhibits the issue

my My issue seems pretty trivial, however, I cannot find a way.

I create a manifold, with 2 frames/charts, and create transition_map between them. them. Create a metric. metric. I create then a submanifold, create diff_map diff_map to the first manifold, manifold, and make an embedding. Then I try to get the induced_metric.induced_metric.

It works only if the diff_map diff_map is referenced to the first created frame/chart at top level.

In my exemple : example:

• top manifold with cartesian and cylindrical charts charts
• sub manifold is a cylinder. I cannot base it on the cylindrical top chart. chart.

My purpose would be to have a top manifold series of charts charts at top and natural embeddings of lower dimension manifolds. manifolds.

Any idea?

My code : code:

E = Manifold(3, 'E', structure='Riemannian')
cartesian.<x,y,z>=E.chart()
cartesian_frame=cartesian.frame()
 
cartesian.<x,y,z> = E.chart()
cartesian_frame = cartesian.frame()

cylindrical.<r,ph,z> = E.chart(r'r:(0,+oo) ph:(0,2pi):\phi ph:(0,2*pi):\phi z')
cylindrical_frame=cylindrical.frame() 
cylindrical_frame = cylindrical.frame()  
cyl2cart = cylindrical.transition_map(cartesian , [rcos(ph),r*sin(ph),z])
[r*cos(ph), r*sin(ph), z])
cart2cyl = cyl2cart.set_inverse(sqrt(x^2+y^2),  atan2(y,x),z)
 
ecc=E.coord_changes()
cyl2cart.set_inverse(sqrt(x^2 + y^2), atan2(y, x), z)

ecc = E.coord_changes()
for cc in ecc :
    display(ecc[cc].display())
 
g=E.metric()
ecc:
    display(ecc[cc].display())

g = E.metric()
g[cartesian_frame, 0,0],g[cartesian_frame,1,1],g[cartesian_frame,2,2]=1,1,1
0, 0], g[cartesian_frame, 1, 1], g[cartesian_frame, 2, 2] = 1, 1, 1
display(g.display(cartesian))
display(g.display(cylindrical))
 
display(g.display(cylindrical))

C2 = Manifold(2, 'C2',ambient=E, 'C2', ambient=E, structure='Riemannian')
Cyl2.<ph,z>=C2.chart(r' ph:\phi Cyl2.<ph,z> = C2.chart(r'ph:\phi z')
Cyl2_frame=Cyl2.frame()
 
Cyl2_frame = Cyl2.frame()

if True :                                                ###  True:  # True shows issue
    E.set_default_frame(cylindrical_frame)
    E.set_default_chart(cylindrical)
    Cyl2tocylindrical=C2.diff_map(E,{(Cyl2,cylindrical):[1,ph,z]})
    cylindricaltoCyl2=E.diff_map(C2,{(cylindrical,Cyl2):[ph,z]})
    C2.set_embedding(Cyl2tocylindrical)    
 
else :
    Cyl2tocartesian=C2.diff_map(E,{(Cyl2,cartesian):[cos(ph),sin(ph),z]})
    cartesiantoCyl2=E.diff_map(C2,{(cartesian,Cyl2):[arctan2(y,x),z]})
    C2.set_embedding(Cyl2tocartesian)
 
C2g=C2.induced_metric()                                  ### Never Cyl2tocylindrical = C2.diff_map(E, {(Cyl2, cylindrical): [1, ph, z]})
    cylindricaltoCyl2 = E.diff_map(C2, {(cylindrical, Cyl2): [ph, z]})
    C2.set_embedding(Cyl2tocylindrical)

else:
    Cyl2tocartesian = C2.diff_map(E, {(Cyl2, cartesian): [cos(ph), sin(ph), z]})
    cartesiantoCyl2 = E.diff_map(C2, {(cartesian, Cyl2): [arctan2(y, x), z]})
    C2.set_embedding(Cyl2tocartesian)

C2g = C2.induced_metric()  # never an error message here
display(C2g.display())                                   ### # exhibits the issueissue

My issue seems pretty trivial, however, I cannot find a way.

I create a manifold, with 2 frames/charts, and create transition_map between them. Create a metric. I create then a submanifold, create diff_map to the first manifold, and make an embedding. Then I try to get the induced_metric.

It works only if the diff_map is referenced to the first created frame/chart at top level.

In my example:

• top manifold with cartesian and cylindrical charts
• sub manifold is a cylinder. I cannot base it on the cylindrical top chart.

My purpose would be to have a top manifold series of charts at top and natural embeddings of lower dimension manifolds.

Any idea?

My code:

E = Manifold(3, 'E', structure='Riemannian')
cartesian.<x,y,z> = E.chart()
cartesian_frame = cartesian.frame()

cylindrical.<r,ph,z> = E.chart(r'r:(0,+oo) ph:(0,2*pi):\phi z')
cylindrical_frame = cylindrical.frame()  
cyl2cart = cylindrical.transition_map(cartesian , [r*cos(ph), r*sin(ph), z])
cart2cyl = cyl2cart.set_inverse(sqrt(x^2 + y^2), atan2(y, x), z)

ecc = E.coord_changes()
for cc in ecc:
    display(ecc[cc].display())

g = E.metric()
g[cartesian_frame, 0, 0], g[cartesian_frame, 1, 1], g[cartesian_frame, 2, 2] = 1, 1, 1
display(g.display(cartesian))
display(g.display(cylindrical))

C2 = Manifold(2, 'C2', ambient=E, structure='Riemannian')
Cyl2.<ph,z> = C2.chart(r'ph:\phi z')
Cyl2_frame = Cyl2.frame()

if True:  # True shows issue
    E.set_default_frame(cylindrical_frame)
    E.set_default_chart(cylindrical)
    Cyl2tocylindrical = C2.diff_map(E, {(Cyl2, cylindrical): [1, ph, z]})
    cylindricaltoCyl2 = E.diff_map(C2, {(cylindrical, Cyl2): [ph, z]})
    C2.set_embedding(Cyl2tocylindrical)

else:
    Cyl2tocartesian = C2.diff_map(E, {(Cyl2, cartesian): [cos(ph), sin(ph), z]})
    cartesiantoCyl2 = E.diff_map(C2, {(cartesian, Cyl2): [arctan2(y, x), z]})
    C2.set_embedding(Cyl2tocartesian)

C2g = C2.induced_metric()  # never an error message here
display(C2g.display())  # exhibits the issue

My issue seems pretty trivial, however, I cannot find a way.

I create a manifold, with 2 frames/charts, and create transition_map between them. Create a metric. I create then a submanifold, create diff_map to the first manifold, and make an embedding. Then I try to get the induced_metric.

It works only if the diff_map is referenced to the first created frame/chart at top level.

In my example:

• top manifold with cartesian and cylindrical charts
• sub manifold is a cylinder. I cannot base it on the cylindrical top chart.

My purpose would be to have a top manifold series of charts at top and natural embeddings of lower dimension manifolds.

Any idea?

My code:

E = Manifold(3, 'E', structure='Riemannian')
cartesian.<x,y,z> = E.chart()
cartesian_frame = cartesian.frame()

cylindrical.<r,ph,z> = E.chart(r'r:(0,+oo) ph:(0,2*pi):\phi z')
cylindrical_frame = cylindrical.frame()  
cyl2cart = cylindrical.transition_map(cartesian , [r*cos(ph), r*sin(ph), z])
cart2cyl = cyl2cart.set_inverse(sqrt(x^2 + y^2), atan2(y, x), z)

ecc = E.coord_changes()
for cc in ecc:
    display(ecc[cc].display())

g = E.metric()
g[cartesian_frame, 0, 0], g[cartesian_frame, 1, 1], g[cartesian_frame, 2, 2] = 1, 1, 1
display(g.display(cartesian))
display(g.display(cylindrical))

C2 = Manifold(2, 'C2', ambient=E, structure='Riemannian')
Cyl2.<ph,z> = C2.chart(r'ph:\phi z')
Cyl2_frame = Cyl2.frame()

if True:  # True shows issue
    E.set_default_frame(cylindrical_frame)
    E.set_default_chart(cylindrical)
    Cyl2tocylindrical = C2.diff_map(E, {(Cyl2, cylindrical): [1, ph, z]})
    cylindricaltoCyl2 = E.diff_map(C2, {(cylindrical, Cyl2): [ph, z]})
    C2.set_embedding(Cyl2tocylindrical)

else:
    Cyl2tocartesian = C2.diff_map(E, {(Cyl2, cartesian): [cos(ph), sin(ph), z]})
    cartesiantoCyl2 = E.diff_map(C2, {(cartesian, Cyl2): [arctan2(y, x), z]})
    C2.set_embedding(Cyl2tocartesian)

C2g = C2.induced_metric()  # never an error message here
display(C2g.display())  # exhibits the issue

My issue seems pretty trivial, however, I cannot find a way.

I create a manifold, with 2 frames/charts, and create transition_map between them. Create a metric. I create then a submanifold, create diff_map to the first manifold, and make an embedding. Then I try to get the induced_metric.

It works only if the diff_map is referenced to the first created frame/chart at top level.

In my example:

• top manifold with cartesian and cylindrical charts
• sub manifold is a cylinder. I cannot base it on the cylindrical top chart.

My purpose would be to have a top manifold series of charts at top and natural embeddings of lower dimension manifolds.

Any idea?

My code:

E = Manifold(3, 'E', structure='Riemannian')
cartesian.<x,y,z> = E.chart()
cartesian_frame = cartesian.frame()

cylindrical.<r,ph,z> = E.chart(r'r:(0,+oo) ph:(0,2*pi):\phi z')
cylindrical_frame = cylindrical.frame()  
cyl2cart = cylindrical.transition_map(cartesian , [r*cos(ph), r*sin(ph), z])
cart2cyl = cyl2cart.set_inverse(sqrt(x^2 + y^2), atan2(y, x), z)

ecc = E.coord_changes()
for cc in ecc:
    display(ecc[cc].display())

g = E.metric()
g[cartesian_frame, 0, 0], g[cartesian_frame, 1, 1], g[cartesian_frame, 2, 2] = 1, 1, 1
display(g.display(cartesian))
display(g.display(cylindrical))

C2 = Manifold(2, 'C2', ambient=E, structure='Riemannian')
Cyl2.<ph,z> = C2.chart(r'ph:\phi z')
Cyl2_frame = Cyl2.frame()

if True:  # True shows issue
    E.set_default_frame(cylindrical_frame)
    E.set_default_chart(cylindrical)
    Cyl2tocylindrical = C2.diff_map(E, {(Cyl2, cylindrical): [1, ph, z]})
    cylindricaltoCyl2 = E.diff_map(C2, {(cylindrical, Cyl2): [ph, z]})
    C2.set_embedding(Cyl2tocylindrical)

else:
    Cyl2tocartesian = C2.diff_map(E, {(Cyl2, cartesian): [cos(ph), sin(ph), z]})
    cartesiantoCyl2 = E.diff_map(C2, {(cartesian, Cyl2): [arctan2(y, x), z]})
    C2.set_embedding(Cyl2tocartesian)

C2g = C2.induced_metric()  # never an error message here
display(C2g.display())  # exhibits the issue