On a geodesic mapping of a square

Let $$X$$ be a proper geodesic space which is uniquely geodesic. Let $$phi:[0,1]times[0,1] to X$$ be a function which satisfies the following:

The maps $$phi(0,cdot)$$, $$phi(cdot,0)$$, $$phi(1,cdot)$$, and $$phi(cdot,1)$$ are all (linearly parametrized) geodesics. Furthermore, for each fixed $$s$$, the map $$phi(s,cdot)$$ is a (linearly parametrized) geodesic connecting $$phi(s,0)$$ to $$phi(s,1)$$.

Given the above conditions, is it true that that for any fixed $$t$$, the map $$phi(cdot,t)$$ is a geodesic connecting $$phi(0,t)$$ to $$phi(1,t)$$? If not, is there a condition we can apply for which this is true (e.g. the space must be Hadamard)?

MathOverflow Asked by Logan Fox on August 8, 2020

This is not true. Let $$X$$ be the unit sphere, or some hemisphere thereof, which we describe first in spherical coordinates.

Let $$f(s,0)$$ go east along the equator, $$(theta,phi)=(2spi/3,pi/2)$$.

Let $$f(s,1)$$ go south from the North Pole, $$(theta,phi)=(pi,spi/3)$$

Let $$f(s,t)$$ be $$t$$ of the way from $$f(s,0)$$ to $$f(s,1)$$.

Then $$f(s,1/2)$$ is not a geodesic.

Each $$f(s,1/2)$$ is the midpoint of $$f(s,0)$$ and $$f(s,1)$$, so it is proportional to $$f(s,0)+f(s,1)$$ in $$mathbb{R}^3$$. Thus in Cartesian coordinates:

begin{align} fleft(0,frac12right) propto, & big(phantom{-sqrt{3}},1phantom{sqrt{3}}, 0 , 1 big) \ fleft(frac12,frac12right) propto, & left(phantom{-sqrt{3}}0phantom{sqrt{3}}, frac{sqrt{3}}2, frac{sqrt{3}}2right)\ fleft(1,frac12right) propto, & left(frac{-1-sqrt{3}}2, frac{sqrt{3}}2, frac12 right) end{align} These three vectors have non-zero determinant, so they are not in the same plane through the origin, and $$f(1/2,1/2)$$ is not on the geodesic between the other two.

Correct answer by Matt F. on August 8, 2020

Related Questions

triviality of homology with local coefficients

1  Asked on February 17, 2021

Generators of sandpile groups of wheel graphs

1  Asked on February 16, 2021 by castor

Conditions for pointwise convergence of indicators precomposed with uniformly continuous sequence

1  Asked on February 16, 2021 by bernard_karkanidis

(Weakly) connected sets with large (out-)boundary

0  Asked on February 15, 2021

Global minimum of sum of a non-convex and convex function, where minima of the non-convex function can be found

0  Asked on February 15, 2021 by proof-by-wine

Hilbert class field of Quadratic fields

1  Asked on February 14, 2021 by sina

The variety induced by an extension of a field

2  Asked on February 12, 2021 by federico-fallucca

Why is this algebra called the q-Weyl algebra?

1  Asked on February 12, 2021 by gmra

Cohomology of a simplicial abelian group $X_bullet$, where $S_n$ acts on $X_n$

1  Asked on February 12, 2021 by patrick-nicodemus

Hyperplane arrangements whose regions all have the same shape

1  Asked on February 12, 2021 by christian-gaetz

Twisted winding number

1  Asked on February 9, 2021 by jack-l

Generalized moment problem for discrete distributions

1  Asked on February 8, 2021 by harharkh

Convergence of Riemann’s Product representation of Xi

2  Asked on February 7, 2021 by mustafa-said

Positive subharmonic functions with constant integral blowing up at boundary

1  Asked on February 7, 2021 by fozz

Symmetry in Hardy-Littlewood k-tuple conjecture

1  Asked on February 6, 2021 by sylvain-julien

Looking up the Mordell-Weil rank and generator(s) of a Weierstrass Equation

4  Asked on February 5, 2021 by john-r-ramsden

A question on moduli space of Hitchin’s equations

1  Asked on February 5, 2021

Writing papers in pre-LaTeX era?

28  Asked on February 3, 2021 by psihodelia

Checking if Hochschild cohomology $mathit{HH}^2(A)=0$

2  Asked on February 2, 2021 by serge

On a degenerate SDE in the unit ball

0  Asked on February 2, 2021