diff --git a/src/sage/manifolds/differentiable/chart.py b/src/sage/manifolds/differentiable/chart.py
index a2c40d33231..70c6b885253 100644
--- a/src/sage/manifolds/differentiable/chart.py
+++ b/src/sage/manifolds/differentiable/chart.py
@@ -488,7 +488,7 @@ def coframe(self):
             sage: ey = c_xy.frame()[1] ; ey
             Vector field d/dy on the 2-dimensional differentiable manifold M
             sage: dx(ex).display()
-             dx(d/dx): M --> R
+            dx(d/dx): M --> R
                (x, y) |--> 1
             sage: dx(ey).display()
             zero: M --> R
diff --git a/src/sage/manifolds/differentiable/tensorfield.py b/src/sage/manifolds/differentiable/tensorfield.py
index 654096b41ae..57fb7f24250 100644
--- a/src/sage/manifolds/differentiable/tensorfield.py
+++ b/src/sage/manifolds/differentiable/tensorfield.py
@@ -4538,7 +4538,7 @@ def apply_map(self, fun, frame=None, chart=None,
             sage: v.display(P.frame(), P)
             (p^2 - q^2 - 2*p - 2*q) d/dp + (2*pi - 2*pi^2 - 2*(pi - pi^2)*p) d/dq
             sage: v.display(P.frame(), X)
-            (x^2 - y^2) d/dp - 2*(pi - pi^2)*x d/dq
+            (x + y)*(x - y) d/dp + 2*pi*(pi - 1)*x d/dq
 
         By default, the components of ``v`` in frames distinct from the
         specified one have been deleted::
@@ -4553,7 +4553,7 @@ def apply_map(self, fun, frame=None, chart=None,
         is effective in ``X.frame()`` as well::
 
             sage: v.display(X.frame(), X)
-            (x^2 - y^2) d/dx - 2*(pi - pi^2)*x d/dy
+            (x + y)*(x - y) d/dx + 2*pi*(pi - 1)*x d/dy
 
         When the requested operation does not change the value of the tensor
         field, one can use the keyword argument ``keep_other_components=True``,
diff --git a/src/sage/manifolds/local_frame.py b/src/sage/manifolds/local_frame.py
index 507c37055c6..7830e24a280 100644
--- a/src/sage/manifolds/local_frame.py
+++ b/src/sage/manifolds/local_frame.py
@@ -119,7 +119,7 @@
 Let us check that the coframe `(e^i)` is indeed the dual of the vector
 frame `(e_i)`::
 
-    sage: e_dual[1](e[1]) # the linear form e^1 applied to the local section e_1
+    sage: e_dual[1](e[1]) # linear form e^1 applied to local section e_1
     Scalar field e^1(e_1) on the Open subset U of the 3-dimensional topological
      manifold M
     sage: e_dual[1](e[1]).expr() # the explicit expression of e^1(e_1)
diff --git a/src/sage/manifolds/scalarfield.py b/src/sage/manifolds/scalarfield.py
index e48520c1ed6..38514061dbb 100644
--- a/src/sage/manifolds/scalarfield.py
+++ b/src/sage/manifolds/scalarfield.py
@@ -2508,10 +2508,10 @@ def _add_(self, other):
             True
 
         """
-        # Special cases:
-        if self._is_zero:
+        # Trivial cases:
+        if self.is_trivial_zero():
             return other
-        if other._is_zero:
+        if other.is_trivial_zero():
             return self
         # Generic case:
         com_charts = self.common_charts(other)
@@ -2557,11 +2557,13 @@ def _sub_(self, other):
             True
 
         """
-        # Special cases:
-        if self._is_zero:
+        # Trivial cases:
+        if self.is_trivial_zero():
             return -other
-        if other._is_zero:
+        if other.is_trivial_zero():
             return self
+        if self is other:
+            return self.parent().zero()
         # Generic case:
         com_charts = self.common_charts(other)
         if com_charts is None:
@@ -2576,7 +2578,6 @@ def _sub_(self, other):
             result._latex_name = self._latex_name + '-' + other._latex_name
         return result
 
-
     def _mul_(self, other):
         r"""
         Scalar field multiplication.
@@ -2609,12 +2610,16 @@ def _mul_(self, other):
             True
 
         """
-        from sage.tensor.modules.format_utilities import (format_mul_txt,
-                                                         format_mul_latex)
-        # Special cases:
-        if self._is_zero or other._is_zero:
+        # Trivial cases:
+        if self.is_trivial_zero() or other.is_trivial_zero():
             return self._domain.zero_scalar_field()
+        if (self - 1).is_trivial_zero():
+            return other
+        if (other - 1).is_trivial_zero():
+            return self
         # Generic case:
+        from sage.tensor.modules.format_utilities import (format_mul_txt,
+                                                          format_mul_latex)
         com_charts = self.common_charts(other)
         if com_charts is None:
             raise ValueError("no common chart for the multiplication")
@@ -2661,10 +2666,10 @@ def _div_(self, other):
         """
         from sage.tensor.modules.format_utilities import format_mul_txt, \
                                                          format_mul_latex
-        # Special cases:
-        if other._is_zero:
+        # Trivial cases:
+        if other.is_trivial_zero():
             raise ZeroDivisionError("division of a scalar field by zero")
-        if self._is_zero:
+        if self.is_trivial_zero():
             return self._domain.zero_scalar_field()
         # Generic case:
         com_charts = self.common_charts(other)
@@ -2726,6 +2731,7 @@ def _lmul_(self, number):
             True
 
         """
+        # Trivial cases:
         try:
             if number.is_trivial_zero():
                 return self.parent().zero()
@@ -2737,6 +2743,7 @@ def _lmul_(self, number):
                 return self.parent().zero()
             if number == 1:
                 return self
+        # Generic case:
         result = type(self)(self.parent())
         if isinstance(number, Expression):
             var = number.variables()  # possible symbolic variables in number