What characteristic lines on the pseudosphere can form a triangle whose internal angle sum is $180^\circ$?
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If by "characteristic lines" you mean "geodesics", then this does not happen. All triangles have angle sum strictly less than $180^\circ$, by the Gauss-Bonnet theorem.
Added: Your comment seems to indicate that you are asking not about "geodesics" but simply about arbitrary smooth curves. If so, then there are almost no restrictions on the angle sum whatsoever. If I start with any triangle having three smooth curves for its sides, meeting at three angles, then I can alter the sides of the triangles by arbitrarily small amounts near each of the three angles ("small" meaning $C^0$) to make the each internal angle be anything in the interval $(0,2\pi)$ and hence their sum can be anything in the interval $(0,6\pi)$.
Lee Mosher
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Thanks for comment. There are several types of lines that can be drawn. Question is, if not a geodesic then what is its differential equation, what kind or type is the curved line? – Narasimham May 29 '16 at 17:07
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I've added something to my answer in an attempt to address your comment. Unless you have some specific type of "line" in mind, I doubt anything more can be said. – Lee Mosher May 29 '16 at 17:43
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(+1) Presumably the question complements Triangle of zero spherical excess on a sphere. – Andrew D. Hwang May 29 '16 at 18:15
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I posed the question as in principle it makes sense, (to me at least), such a condition is feasible on both the types. – Narasimham May 29 '16 at 18:55
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@LeeMosher. With geodesics we have a pseudo spherical deficit. So the the question is how should the lines $ uniformly $ and differentially be altered so that the geodesic curvature integral sums up in the required way. – Narasimham May 29 '16 at 19:32
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You mention geodesic curvature integrals... are you familiar with the Gauss-Bonnet theorem? – Lee Mosher May 29 '16 at 23:50