Mixing
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Sign of geodesic curvature in GB Theorem
$begingroup$
With the usual notation we have $chi(M)=1 $ let us consider for a spherical cap ( compact surface with boundary) the GB theorem in $ mathbb R^3$:
$$ int_M K dA + int_ { partial M}kappa_{g} ,ds= 2 pi, chi(M) = 2 pi $$
For a polar cap when a polar latitude circle is drawn infinitesimally near North Pole, the respective terms are
$$ 0 + 2pi =2 pi $$
For a hemisphere cap when an equator /great circle is drawn in the center
$$ 2pi +0 = 2 pi$$
For a polar cap when a polar latitude circle is drawn infinitesimally near South Pole enclosing all of the sphere surface
$$ 4pi -2 pi =2 pi$$
Now considering same differential geometric definition of the same circle how do we obtain $ int kappa_{g} ,ds = pm 2 pi$ for North pole or South Pole? $kappa_g$ is a signed scalar.. so how is it reckoned independently?
differential-geometry
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
$endgroup$
add a comment |
$begingroup$
With the usual notation we have $chi(M)=1 $ let us consider for a spherical cap ( compact surface with boundary) the GB theorem in $ mathbb R^3$:
$$ int_M K dA + int_ { partial M}kappa_{g} ,ds= 2 pi, chi(M) = 2 pi $$
For a polar cap when a polar latitude circle is drawn infinitesimally near North Pole, the respective terms are
$$ 0 + 2pi =2 pi $$
For a hemisphere cap when an equator /great circle is drawn in the center
$$ 2pi +0 = 2 pi$$
For a polar cap when a polar latitude circle is drawn infinitesimally near South Pole enclosing all of the sphere surface
$$ 4pi -2 pi =2 pi$$
Now considering same differential geometric definition of the same circle how do we obtain $ int kappa_{g} ,ds = pm 2 pi$ for North pole or South Pole? $kappa_g$ is a signed scalar.. so how is it reckoned independently?
differential-geometry
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
$endgroup$
add a comment |
$begingroup$
With the usual notation we have $chi(M)=1 $ let us consider for a spherical cap ( compact surface with boundary) the GB theorem in $ mathbb R^3$:
$$ int_M K dA + int_ { partial M}kappa_{g} ,ds= 2 pi, chi(M) = 2 pi $$
For a polar cap when a polar latitude circle is drawn infinitesimally near North Pole, the respective terms are
$$ 0 + 2pi =2 pi $$
For a hemisphere cap when an equator /great circle is drawn in the center
$$ 2pi +0 = 2 pi$$
For a polar cap when a polar latitude circle is drawn infinitesimally near South Pole enclosing all of the sphere surface
$$ 4pi -2 pi =2 pi$$
Now considering same differential geometric definition of the same circle how do we obtain $ int kappa_{g} ,ds = pm 2 pi$ for North pole or South Pole? $kappa_g$ is a signed scalar.. so how is it reckoned independently?
differential-geometry
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
$endgroup$
With the usual notation we have $chi(M)=1 $ let us consider for a spherical cap ( compact surface with boundary) the GB theorem in $ mathbb R^3$:
$$ int_M K dA + int_ { partial M}kappa_{g} ,ds= 2 pi, chi(M) = 2 pi $$
For a polar cap when a polar latitude circle is drawn infinitesimally near North Pole, the respective terms are
$$ 0 + 2pi =2 pi $$
For a hemisphere cap when an equator /great circle is drawn in the center
$$ 2pi +0 = 2 pi$$
For a polar cap when a polar latitude circle is drawn infinitesimally near South Pole enclosing all of the sphere surface
$$ 4pi -2 pi =2 pi$$
Now considering same differential geometric definition of the same circle how do we obtain $ int kappa_{g} ,ds = pm 2 pi$ for North pole or South Pole? $kappa_g$ is a signed scalar.. so how is it reckoned independently?
differential-geometry
differential-geometry
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
edited Jan 12 at 7:16
Narasimham
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
asked Jan 7 at 18:37
NarasimhamNarasimham
20.7k52158
20.7k52158
add a comment |
add a comment |
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