Abstract
We want to study the metric of a Riemannian manifold. The first tasks to address are:
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1.
to compute the metric d as defined by equation 4.13 on page 174 (namely itd (p, q) is the infimum of the lengths of curves connecting p to q)
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2.
to determine if there are curves realizing this distance (called segments or shortest paths or minimal geodesics according to your taste) and
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3.
to study them.
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References
The gradient vector field ∇f is defined to be the vector field dual to the differential df (which is a 1-form) in the sense that \( g\left( {\nabla f,v} \right) = df\left( v \right) \) for any tangent vector v.
Rauch had a statement for curves joining the extremities of the triangle.
See on page 268 for the definition of conjugate point.
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© 2003 Springer-Verlag Berlin Heidelberg
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Berger, M. (2003). Riemannian Manifolds as Metric Spaces and the Geometric Meaning of Sectional and Ricci Curvature. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18245-7_6
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DOI: https://doi.org/10.1007/978-3-642-18245-7_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65317-2
Online ISBN: 978-3-642-18245-7
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