Gauss-bonnet just for geodesic triangles
$begingroup$
I have a question about some ways for proving the Gauss-Bonnet theorem just for small geodesic triangles. The general formula for the Gauss-Bonnet theorem is
$$iint_R KdS+sum_{i=0}^kint_{s_i}^{s_{i+1}} k_gds+sum_{i=0}^ktheta_i=2pi.$$
The ingredients here are a small portion $R$ of a surface $S$, its boundary constituted by $k$ arcs (not necessarily geodesic arcs) and the ''exterior'' angles $theta_i$ measured counterclockwise at the corner of the mentioned boundary, $K$ is the gaussian curvature of the surface and $k_g$ is the geodesic curvature of the portions of the boundary of $R$.
Of course, if we consider the region $R$ to be a small geodesic triangle $T$, that is, it is contained in some small normal neighborhood of the surface and its boundary is the union of three small geodesic segments, we have the following nice version of Gauss-Bonnet formula
$$iint_T KdS=2pi-theta_1-theta_2-theta_3,$$
where the $theta_i$ are just the external angles at the corners of the geodesic boundary triangle of $T$. My question is, then: is there an elementary way to obtaining the "geodesic triangle version" above without using the "complete version" (for example, using Stoke's theorem)?
Any reference will be of great help! Thanks in advance for all the community!
differential-geometry reference-request curvature geodesic stokes-theorem
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
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add a comment |
$begingroup$
I have a question about some ways for proving the Gauss-Bonnet theorem just for small geodesic triangles. The general formula for the Gauss-Bonnet theorem is
$$iint_R KdS+sum_{i=0}^kint_{s_i}^{s_{i+1}} k_gds+sum_{i=0}^ktheta_i=2pi.$$
The ingredients here are a small portion $R$ of a surface $S$, its boundary constituted by $k$ arcs (not necessarily geodesic arcs) and the ''exterior'' angles $theta_i$ measured counterclockwise at the corner of the mentioned boundary, $K$ is the gaussian curvature of the surface and $k_g$ is the geodesic curvature of the portions of the boundary of $R$.
Of course, if we consider the region $R$ to be a small geodesic triangle $T$, that is, it is contained in some small normal neighborhood of the surface and its boundary is the union of three small geodesic segments, we have the following nice version of Gauss-Bonnet formula
$$iint_T KdS=2pi-theta_1-theta_2-theta_3,$$
where the $theta_i$ are just the external angles at the corners of the geodesic boundary triangle of $T$. My question is, then: is there an elementary way to obtaining the "geodesic triangle version" above without using the "complete version" (for example, using Stoke's theorem)?
Any reference will be of great help! Thanks in advance for all the community!
differential-geometry reference-request curvature geodesic stokes-theorem
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
$endgroup$
1
$begingroup$
But the original Local Gauss Bonnet Theorem is just an immediate application of Stokes's Theorem (especially if you know differential forms). See p.~105 of my text. [Unfortunately, the university server is messed up, but I hope it'll be fixed soon.] At any rate, I can give easier proofs in the case of constant curvature, but certainly none in the general case.
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:12
$begingroup$
Dear @TedShifrin, the case is not using differential forms. I am hoping to show this to some students soon. They have no experience with differential forms. Of course I can "translate" the language. With luck, the answer to my question could be sort of adapted to their language already, but yours is already of great help!
$endgroup$
– matgaio
Nov 25 '18 at 23:21
1
$begingroup$
No, I know no elementary argument for the geodesic triangle case in general. Sorry. :(
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:27
add a comment |
$begingroup$
I have a question about some ways for proving the Gauss-Bonnet theorem just for small geodesic triangles. The general formula for the Gauss-Bonnet theorem is
$$iint_R KdS+sum_{i=0}^kint_{s_i}^{s_{i+1}} k_gds+sum_{i=0}^ktheta_i=2pi.$$
The ingredients here are a small portion $R$ of a surface $S$, its boundary constituted by $k$ arcs (not necessarily geodesic arcs) and the ''exterior'' angles $theta_i$ measured counterclockwise at the corner of the mentioned boundary, $K$ is the gaussian curvature of the surface and $k_g$ is the geodesic curvature of the portions of the boundary of $R$.
Of course, if we consider the region $R$ to be a small geodesic triangle $T$, that is, it is contained in some small normal neighborhood of the surface and its boundary is the union of three small geodesic segments, we have the following nice version of Gauss-Bonnet formula
$$iint_T KdS=2pi-theta_1-theta_2-theta_3,$$
where the $theta_i$ are just the external angles at the corners of the geodesic boundary triangle of $T$. My question is, then: is there an elementary way to obtaining the "geodesic triangle version" above without using the "complete version" (for example, using Stoke's theorem)?
Any reference will be of great help! Thanks in advance for all the community!
differential-geometry reference-request curvature geodesic stokes-theorem
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
$endgroup$
I have a question about some ways for proving the Gauss-Bonnet theorem just for small geodesic triangles. The general formula for the Gauss-Bonnet theorem is
$$iint_R KdS+sum_{i=0}^kint_{s_i}^{s_{i+1}} k_gds+sum_{i=0}^ktheta_i=2pi.$$
The ingredients here are a small portion $R$ of a surface $S$, its boundary constituted by $k$ arcs (not necessarily geodesic arcs) and the ''exterior'' angles $theta_i$ measured counterclockwise at the corner of the mentioned boundary, $K$ is the gaussian curvature of the surface and $k_g$ is the geodesic curvature of the portions of the boundary of $R$.
Of course, if we consider the region $R$ to be a small geodesic triangle $T$, that is, it is contained in some small normal neighborhood of the surface and its boundary is the union of three small geodesic segments, we have the following nice version of Gauss-Bonnet formula
$$iint_T KdS=2pi-theta_1-theta_2-theta_3,$$
where the $theta_i$ are just the external angles at the corners of the geodesic boundary triangle of $T$. My question is, then: is there an elementary way to obtaining the "geodesic triangle version" above without using the "complete version" (for example, using Stoke's theorem)?
Any reference will be of great help! Thanks in advance for all the community!
differential-geometry reference-request curvature geodesic stokes-theorem
differential-geometry reference-request curvature geodesic stokes-theorem
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
asked Nov 25 '18 at 22:49
matgaiomatgaio
1,576816
1,576816
1
$begingroup$
But the original Local Gauss Bonnet Theorem is just an immediate application of Stokes's Theorem (especially if you know differential forms). See p.~105 of my text. [Unfortunately, the university server is messed up, but I hope it'll be fixed soon.] At any rate, I can give easier proofs in the case of constant curvature, but certainly none in the general case.
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:12
$begingroup$
Dear @TedShifrin, the case is not using differential forms. I am hoping to show this to some students soon. They have no experience with differential forms. Of course I can "translate" the language. With luck, the answer to my question could be sort of adapted to their language already, but yours is already of great help!
$endgroup$
– matgaio
Nov 25 '18 at 23:21
1
$begingroup$
No, I know no elementary argument for the geodesic triangle case in general. Sorry. :(
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:27
add a comment |
1
$begingroup$
But the original Local Gauss Bonnet Theorem is just an immediate application of Stokes's Theorem (especially if you know differential forms). See p.~105 of my text. [Unfortunately, the university server is messed up, but I hope it'll be fixed soon.] At any rate, I can give easier proofs in the case of constant curvature, but certainly none in the general case.
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:12
$begingroup$
Dear @TedShifrin, the case is not using differential forms. I am hoping to show this to some students soon. They have no experience with differential forms. Of course I can "translate" the language. With luck, the answer to my question could be sort of adapted to their language already, but yours is already of great help!
$endgroup$
– matgaio
Nov 25 '18 at 23:21
1
$begingroup$
No, I know no elementary argument for the geodesic triangle case in general. Sorry. :(
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:27
1
1
$begingroup$
But the original Local Gauss Bonnet Theorem is just an immediate application of Stokes's Theorem (especially if you know differential forms). See p.~105 of my text. [Unfortunately, the university server is messed up, but I hope it'll be fixed soon.] At any rate, I can give easier proofs in the case of constant curvature, but certainly none in the general case.
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:12
$begingroup$
But the original Local Gauss Bonnet Theorem is just an immediate application of Stokes's Theorem (especially if you know differential forms). See p.~105 of my text. [Unfortunately, the university server is messed up, but I hope it'll be fixed soon.] At any rate, I can give easier proofs in the case of constant curvature, but certainly none in the general case.
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:12
$begingroup$
Dear @TedShifrin, the case is not using differential forms. I am hoping to show this to some students soon. They have no experience with differential forms. Of course I can "translate" the language. With luck, the answer to my question could be sort of adapted to their language already, but yours is already of great help!
$endgroup$
– matgaio
Nov 25 '18 at 23:21
$begingroup$
Dear @TedShifrin, the case is not using differential forms. I am hoping to show this to some students soon. They have no experience with differential forms. Of course I can "translate" the language. With luck, the answer to my question could be sort of adapted to their language already, but yours is already of great help!
$endgroup$
– matgaio
Nov 25 '18 at 23:21
1
1
$begingroup$
No, I know no elementary argument for the geodesic triangle case in general. Sorry. :(
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:27
$begingroup$
No, I know no elementary argument for the geodesic triangle case in general. Sorry. :(
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:27
add a comment |
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$begingroup$
But the original Local Gauss Bonnet Theorem is just an immediate application of Stokes's Theorem (especially if you know differential forms). See p.~105 of my text. [Unfortunately, the university server is messed up, but I hope it'll be fixed soon.] At any rate, I can give easier proofs in the case of constant curvature, but certainly none in the general case.
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:12
$begingroup$
Dear @TedShifrin, the case is not using differential forms. I am hoping to show this to some students soon. They have no experience with differential forms. Of course I can "translate" the language. With luck, the answer to my question could be sort of adapted to their language already, but yours is already of great help!
$endgroup$
– matgaio
Nov 25 '18 at 23:21
1
$begingroup$
No, I know no elementary argument for the geodesic triangle case in general. Sorry. :(
$endgroup$
– Ted Shifrin
Nov 25 '18 at 23:27