#!/usr/bin/env python
# coding: utf-8

# # 15. Pulback of tensor fields
# 
# This notebook is part of the [Introduction to manifolds in SageMath](https://sagemanifolds.obspm.fr/intro_to_manifolds.html) by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).

# In[1]:


version()


# Recall from [notebook 7](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/07Manifold_FunPullb_Curves.ipynb), that every smooth map between two smooth manifolds $\psi : M → N$  defines a pullback map $\psi^* : C^∞ (N ) → C^∞ (M )$
# by
# 
# $$
# \psi^∗ f = f ◦ \psi\quad \mbox{ for any } f ∈ C^∞ (N ).$$

# <br>
# 
# **Example 15.1**
# 
# If we denote by $\psi$ the coordinate change defined by
# $x=r\cos(\phi),\; y=r\sin(\phi)$  on the open set $U=\{(r,\phi): r>0, 0<\phi <2\pi \},$ then for a scalar function $f$ of variables $x,y,$ the pullback
# $\psi^*f$ is the function $(r,\phi)\to f(r\cos(\phi),r\sin(\phi))$, defined on $U$.
# 
# In the special case $f(x,y)=\sqrt{x^2+y^2}$, the pullback 
# $\psi^*f$ is the function on $U$: $(r,\phi)\to r.$

# In[2]:


get_ipython().run_line_magic('display', 'latex')
M = Manifold(2, 'R^2p')   # manifold R^2
U = M.open_subset('U')    # open subset of R2, 
                          # polar coord. on U: 
c_rphi.<r,phi>=U.chart(r'r:(0,oo) phi:(0,2*pi):\phi')
N = Manifold(2, 'R^2c')   # second copy of R^2                         
c_xy.<x,y> = N.chart()    # Cartesian coordinates:
                          # psi: M --> N : polar to Cart.:
psi = U.diff_map(N, (r*cos(phi), r*sin(phi)),name=r'\psi')
                          # f -scalar function on N 
f=N.scalar_field(sqrt(x^2+y^2),name='f') 

fplb=psi.pullback(f)      # psi^*f -scalar func. on M
fplb.disp()               # show pullback psi^*f


# <br>
# 
# ### Pullback of covariant tensor fields
# 
# <br>
# 
# If $\psi: M\to N$ is a smooth map between smooth manifolds, one  can extend the pullback operation   to a map  $\psi^∗ : T^{(0,k)}N  → T^{(0,k)}M $  by
# 
# \begin{equation}
# (ψ^∗ t)_p (u_{1p} , . . . , u_{kp} ) = t_{ψ( p)} (dψ_p u_{1p} , . . . , dψ_p u_{kp} ),
# \label{}\tag{15.1}
# \end{equation}
# 
# for $u_{1p} , . . . , u_{kp} ∈ T_p M,\ \  p ∈ M.$
# From the linearity of $d\psi_p$ it follows the multilinearity of $(\psi^*t)_p$.
# 
# If $f ∈ C^∞ (N )$, then the differential $d f$ of $f$    is a tensor field of type $(0,1)$,
# consequently from (15.1) it follows
# 
# $$(ψ^∗ d f )_p (v_p ) = d f_{ψ( p)} (dψ_p v_p ),$$
# 
# for $v_p ∈ T_p M.$<br>  From the definitions of $df$ and $d\psi$  we have
# 
# $$d f_{ ψ( p)} (dψ_p v_p ) = dψ_p v_p ( f ) = v_p (ψ^∗ f ) = d(ψ^*f )_p (v_p ).$$
# 
# We have checked that for a smooth map $\ \psi : M → N\ $  and $\ f\in C^∞ (N )$
# 
# \begin{equation}
# ψ^∗ d f = d(ψ^∗ f ).
# \label{}\tag{15.2}
# \end{equation}

# <br>
# 
# **Example 15.2**
# 
# Compute the pullback of differential of  the scalar function from Example 15.1.
# 
# First we compute the left hand side of (15.2):

# In[3]:


# continuation
df=f.differential()          # differential of f
dfplb=psi.pullback(df)       # pullback psi^*df
dfplb.disp(basis=c_rphi)     # display in polar coordinates


# Here is the right hand side of (15.2):

# In[4]:


fplb.differential().disp()  # d(psi^*f)


# <br>
# 
# **Example 15.3**
# 
# Compute pullback of a scalar function and its differential for  general mapping between two-dimensional manifolds $M$ and $N$
# 
# $$\psi: M \to N,\quad \psi(u,v)=
# (\psi_1(u,v),\psi_2(u,v)).
# $$
# 

# In[5]:


get_ipython().run_line_magic('display', 'latex')
M = Manifold(2, r'R^2_{uv}')  # manifold M
c_uv.<u,v>=M.chart()          # coordinates u,v
N = Manifold(2, r'R^2_{xy}')  # manifold N
c_xy.<x,y> = N.chart()        # coordinates x,y
                              # first component of psi
psi1=M.scalar_field(function('psi1')(u,v),name=r'\psi1')
                              # second component of psi
psi2=M.scalar_field(function('psi2')(u,v),name=r'\psi2') 
                              # define psi
psi = M.diff_map(N,(psi1.expr(),psi2.expr()) ,name=r'\psi')
                              # scalar function f
f=N.scalar_field(function('f')(x,y),name='f') 
fplb=psi.pullback(f)          # pullback psi^*f
fplb.disp()                   # show pullback psi^*f


# In[6]:


fplb.differential().disp()   # d(psi^*f)


# In[7]:


df=f.differential()          # df
dfplb=psi.pullback(df)       # psi^*(df)
dfplb.disp()                 # show pullback of df


# <br>
# 
# **Example 15.4**
# 
# Compute the pullback of 1-form  $\ \ 𝛼=𝑎_0(𝑥,𝑦)d𝑥+𝑎_1(𝑥,𝑦)d𝑦\ $ under the map $\psi$ from the previous example.

# In[8]:


get_ipython().run_line_magic('display', 'latex')
M = Manifold(2, r'R^2_{uv}')  # manifold M
c_uv.<u,v>=M.chart()          # coordinates u,v
N = Manifold(2, r'R^2_{xy}')  # manifold N
c_xy.<x,y> = N.chart()        # coordinates x,y
                              # first component of psi
psi1=M.scalar_field(function('psi1')(u,v),name=r'\psi1')
                              # second component of psi
psi2=M.scalar_field(function('psi2')(u,v),name=r'\psi2') 
                              # define psi
psi = M.diff_map(N,(psi1.expr(),psi2.expr()) ,name=r'\psi')
                              # scalar function f
f=N.scalar_field(function('f')(x,y),name='f') 

al=N.diff_form(1,name=r'\alpha')       # 1-form al
astr=['a'+str(i) for i in range(2)]    # list of comp. names
af=[N.scalar_field(function(astr[i])(x,y),name=astr[i]) 
         for i in range(2)]            # list of components
al[:]=af                               # define all components
al.disp()                              # show al


# In[9]:


Manifold.options.omit_function_arguments=False
alplb=psi.pullback(al)                 # pullback psi^*al
alplb.disp()                           # show pullback of al


# Shorter result can be obtained omitting functions arguments.

# In[10]:


Manifold.options.omit_function_arguments=True
alplb.disp()


# <br>
# 
# **Example 15.5**
# 
# Compute the pullback of a tensor field of (0,2)-type  
# $f_{00}(𝑥,𝑦)d𝑥⊗d𝑥+𝑓_{01}(𝑥,𝑦)d𝑥⊗d𝑦+𝑓_{10}(𝑥,𝑦)d𝑦⊗d𝑥+𝑓_{11}(𝑥,𝑦)d𝑦⊗d𝑦\ \ $
# on a two dimensional manifold under $\psi$ from Example 15.3.

# In[11]:


# continuation
t = N.tensor_field(0,2, name='t')  # tensor field t of type (0,2)
                                   # list of component names:
def fn(i,j): return 'f'+str(i)+str(j) 
                                   # list of component functions:
f=[[N.scalar_field(function(fn(j,k))(x,y),name=fn(j,k)) 
     for k in range(2)] for j in range(2)]
t[:]=f                             # define all components of t
t.disp()                           # show t


# In[12]:


Manifold.options.omit_function_arguments=True
tplb=psi.pullback(t)               # pullback psi^*t
tplb.display_comp()                # show pullback components


# <br>
# 
# ### Basic properties of pullback
# 
# <br>

# For  a smooth map $\psi: M \to N,\ $ tensor fields $t$ and $s$ on $N$ of type
# $( 0,k)$  and  constants $a,b\in R\ $ we have
# 
# $$
# (ψ^*(at + bs))_p (u_{1p} , . . . , u_{kp} ) = (at + bs)_{ψ( p)}
# (dψ_p u_{1p} , . . . , dψ_p u_{kp} )\\
# = (at_{ψ( p)} + bs_{ψ( p)} )(dψ_p u_{1p} , . . . , dψ_p u_{kp} )\\
# = at_{ψ( p)} (dψ_p u_{1p} , . . . , dψ_p u_{kp} )
# + bs_{ψ( p)} (dψ_p u_{1p} , . . . , dψ_p u_{kp} )\\
# = a(ψ^∗ t)_p (u_{1p} , . . . , u_{kp} ) + b(ψ^∗ s)_p (u_{1p} , . . . , u_{kp} )\\
# = (aψ^∗ t + bψ^∗ s)_p (u_p , . . . , w_p ),$$
# 
# for $u_{1p} , . . . , u_{kp} ∈ T_p M$. Therefore we have the following linearity result.
# 
# Let $ψ : M → N$ be a  smooth
# map.
# If $t$ and $s$ are tensor fields of type
# $( 0,k)$ on $N$ and $a, b ∈ R$, then
# 
# \begin{equation}
# ψ^∗ (at + bs) = aψ^∗ t + bψ^∗ s.
# \label{}\tag{15.3}
# \end{equation}<br>
# 
# For $\psi\ $ as before, $f\in C^\infty(N)\ $ and $t\in T^{(0.k)}N$, we have
# 
# $$
# (ψ^∗ ( f t)_p (u_{1p} , . . . , u_{kp} ) = 
# ( f t)_{ψ( p)} (dψ_p u_{1p} , . . . , dψ_p u_{kp} )\\
# = f( ψ( p)) t_{ψ( p)} (dψ_p u_{1p} , . . . , dψ_p u_{kp} )
# = (ψ^∗ f )( p) (ψ^∗ t)_p (u_{1p} , . . . , u_{kp} )
# =((ψ^∗ f )(ψ^∗ t))_p (u_{1p} , . . . , u_{kp} ).
# $$
# Thus
# \begin{equation}
# ψ^∗ ( f t) = (ψ^∗ f )(ψ^∗ t).
# \label{}\tag{15.4}
# \end{equation}
# <br>
# 
# If $\psi:M\to N\ $ is smooth, $t$ and $s$ are tensor fields on $N$ of types
# $(0,k)$
# and $(0,l)$ respectively,
#   we have
# 
# $$
# ψ^*(t ⊗ s)_p(u_{1p},...,u_{(k+l)p}) = (t ⊗ s)_{ψ( p)} (dψ_p u_{1p} , . . . , dψ_p u_{(k+l)p} )\\
# = t_{ψ( p)} (dψ_p u_{1p} , . . .,dψ_p u_{kp}) s_{ψ( p)} (dψ_p u_{(k+1)p},. . . , dψ_p u_{(k+l)p} )\\
# = (ψ^∗ t)_p (u_{1p} , . . .,u_{kp})(ψ^∗ s)_p (u_{(k+1)p},. . . , u_{(k+l)p} )
# = ((ψ^∗ t) ⊗ (ψ^∗ s))_p (u_{1p} , . . . , u_{(k+l)p} ),
# $$
# 
# so
# 
# \begin{equation}
# ψ^∗ (t ⊗ s) = (ψ^∗ t) ⊗ (ψ^∗ s).
# \label{}\tag{15.5}
# \end{equation}

# If $M_1, M_2, M_3$ are smooth manifolds, $ψ_1 : M_1 → M_2$ and $ψ_2 : M_2 → M_3$ are smooth maps, then
# 
# $$ (ψ_2 ◦ ψ_1 )^∗ t = (ψ_1^∗ ◦ ψ_2^∗ )\;t,\quad \mbox{for } t ∈ T^{(0,k)} M_3.$$
# 
# To prove this relation let us note that
# $$((ψ_2 ◦ ψ_1 )^∗ t)_p (u_{1p} , . . . , u_{kp} )\\
# = t_{(ψ_2 ◦ψ_1 )( p)} (d(ψ_2 ◦ ψ_1 )_p u_{1p} , . . . , 
# d(ψ_2 ◦ ψ_1 )_p u_{kp})\\
# = t_{ψ_2(ψ_1(p))} (dψ_{2ψ_1( p)} (dψ_{1p} u_{1p} ), . . . , 
#                    dψ_{2ψ_1( p)} (dψ_{1p} u_{kp} ))\\
# = (ψ_2^∗ t)_{ψ 1 ( p)} (dψ_{1 p} u_{1p} , . . . , dψ_{1p} u_{kp} )
# = (ψ_1^∗ (ψ_2^∗ t))_p (u_{1p} , . . . , u_{kp} ).
# $$
# 
# <br>
# 
# ### Pullback of covariant tensor fields in components
# 
# <br>
# 
# If $t$ is a tensor field of type $(0,k)$ on $N$, given locally by $t = t_{i_1... i_k} dy^{i_1} ⊗ · · · ⊗dy^{i_k}$, then the pullback of $t$ under $ψ$ is given by
# 
# $$ψ^∗ t = ψ^∗ (t_{i_1... i_k} dy^{i_1} ⊗ · · · ⊗ dy^{i_k} )\\
# = (ψ^∗ t_{i_1... i_k} )(ψ^∗ dy^{i_1} ) ⊗ · · · ⊗ (ψ^∗ dy^{i_k} )\\
# = (ψ^∗ t_{i_1... i_k} ) d(ψ^∗ y^{i_1} ) ⊗ · · · ⊗ d(ψ^∗ y^{i_k} ).$$
# 
# Since $d(ψ^∗ y^i ) = (∂(ψ^∗ y^i )/∂ x^m) dx^m$,
# where $(x^1 , . . . , x^n )$ are coordinates on $M$, we have
# 
# \begin{equation}
# ψ^∗ t = (ψ^∗ t_{i_1... i_k} )\frac{\partial(\psi^*y^{i_1})}{
# \partial x^{m_1}}\ldots
# \frac{\partial(\psi^*y^{i_k})}{\partial x^{m_k}}dx^{m_1}⊗ · · · ⊗ dx^{m_k}.
# \tag{15.6}
# \end{equation}

# 
# The same formula can be written:
# $$ψ^∗ (t_{i_1... i_k} dy^{i_1} ⊗ · · · ⊗dy^{i_k} )= (t_{i_1... i_k}\circ\psi )\frac{\partial(y^{i_1}\circ\psi)}{
# \partial x^{m_1}}\ldots
# \frac{\partial(y^{i_k}\circ\psi)}{\partial x^{m_k}}dx^{m_1}⊗ · · · ⊗ dx^{m_k}.$$<br>
# So to obtain $\psi^*t$, just replace in $t=t_{i_1... i_k} dy^{i_1} ⊗ · · · ⊗dy^{i_k}$<br>
# $t_{i_1... i_k}\to t_{i_1... i_k}\circ \psi\quad$
# and $dy^i\to \frac{\partial(y^{i}\circ\psi)}{\partial x^{m}}dx^{m}.$

# <br>
# 
# **Example 15.6**
# 
# Consider the covariant tensor field  $ \ g=dx\otimes dx+ dy\otimes dy+ dz\otimes dz$ in $R^3$ and its pullback under the embedding $\psi: S^2\to R^3$ defined by  
# 
# $$ \psi(\theta,\phi)=(\sin \theta \cos \phi, \sin \theta\sin \phi,\cos\theta).$$

# In[13]:


get_ipython().run_line_magic('display', 'latex')
S2=manifolds.Sphere(2)     # manifolds.Sphere module contains the def.
Phi=S2.embedding()                      # of the embedding S^2 -> E^3 
sph.<th,ph>=S2.spherical_coordinates()  # spherical coordinates
E=S2.ambient()                          # 3-dim Euclidean space
c_cart.<x,y,z> = E.cartesian_coordinates()  # Cartesian coord. in E^3
Phi.disp()                              # show Phi


# `metric` is a predefined  covariant tensor field of type (0,2) in $E^3$ 

# In[14]:


g=E.metric()              # g is predefined tensor field of type (0,2)
g.disp()                  # show g


# The pullback of g:

# In[15]:


Phi.pullback(g).disp()                 # pullback of g


# <br>
# 
# **Example 15.7**
# 
# We can repeat the previous  example in higher dimensions. For example we can compute the pullback of $ \ g=dx_1\otimes dx_1+ dx_2\otimes dx_2+ dx_3\otimes dx_3+dx_4\otimes dx_4$
# under the embedding $S^3\to E^4.$

# In[16]:


get_ipython().run_line_magic('display', 'latex')
S3=manifolds.Sphere(3)          # manifolds.Sphere module contains the def.
Phi=S3.embedding()                             # of the embedding S^3 -> E^4
sph.<chi,theta,phi>=S3.spherical_coordinates() # and spherical coordinates
E=S3.ambient()                                 # Euclidean space E^4
c_cart.<x1,x2,x3,x4> = E.cartesian_coordinates() # Cartesian coord. in E^4
fun=Phi.coord_functions()[:];fun   # functions defining the emb. S^3 -> E^4


# `metric` is a predefined  covariant tensor field of type (0,2) in $E^4.$ 

# In[17]:


g=E.metric()                   # g is a predefined (0,2) type tensor field
g.disp()                       # show g


# Pullback of g:

# In[18]:


Phi.pullback(g).disp()        # pullback Psi^*g


# <br>
# 
# 
# Since the differential k-forms are special cases of covariant tensor fields, the pullback operation can be applied to such forms.
# 
# <br>

# **Example 15.8**
# 
# Consider the same embedding $\psi: S^2\to R^3$ as in Example 15.6,  defined by  $$ \psi(\theta,\phi)=(\sin \theta \cos \phi, \sin \theta\sin \phi,\cos\theta),$$ and compute the pullbacks
# $\psi^*(dx\wedge dy),\ \psi^*(dy\wedge dz),\ \psi^*(dx\wedge dz).$

# In[19]:


get_ipython().run_line_magic('display', 'latex')
M = Manifold(3, 'R^3')            # manifold M=R^3
c_xyz.<x,y,z> = M.chart()         # Cartesian coordinates
N = Manifold(2, 'N')              # manifold N=S^2
c_sph.<theta,phi>=N.chart()       # spherical coordinates
                                  # embedding S^2->R^3
psi = N.diff_map(M, (sin(theta)*cos(phi), sin(theta)*sin(phi),cos(theta)),
                 name='psi',latex_name=r'\psi')
E=c_xyz.coframe()                 # coframe (dx,dy,dz)
dx,dy,dz=E[:]                     # define dx,dy,dz
omega1=dx.wedge(dy)               # wedge prod. dx/\dy
omega2=dy.wedge(dz)               # wedge prod. dy/\dz
omega3=dx.wedge(dz)               # wedge prod. dx/\dz
plb=psi.pullback(omega1)          # pullback psi^*omega1
plb.disp()


# In[20]:


plb=psi.pullback(omega2)          # pullback psi^*omega2
plb.disp()


# In[21]:


plb=psi.pullback(omega3)          # pullback psi^*omega3
plb.disp()


# More examples of pullbacks of k-forms can be found in the [next notebook](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/16Manifold_Ext_Der.ipynb).

# ## What's next?
# 
# Take a look at the notebook [Exterior derivative](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/16Manifold_Ext_Der.ipynb).