Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of all equivalence classes of measurable functions $g:X\rightarrow M$ with finite the distance $D(m_0,g)$ from the constant function $m_0$ where $D$ is the metric: $$ D(g,h):=\int_{m \in M} d_g^2(g(m),h(m))d\mu(m) = \int_{m \in M} \|\operatorname{exp}_{g(m)}^{-1}(h(m))\|^2d\mu(m). $$

Fix some non-constant $f \in L^2(\mu;M,m_0)$.

Let $c\in X$ and let $C\subseteq L^2(\mu;M,m_0)$ be proper and non-empty. Under what joint conditions on $C$ and $c$ can we guarantee that $$ \operatorname{argmin}_{g \in C} D(f,g) = \operatorname{argmin}_{g \in C} \int_{m \in M} \|\exp_{c}^{-1}(f(m))-\operatorname{exp}_{c}^{-1}(g(m))\|^2d\mu(m) . $$

Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of all equivalence classes of measurable functions $g:X\rightarrow M$ with finite the distance $D(m_0,g)$ from the constant function $m_0$ where $D$ is the metric: $$ D(g,h):=\int_{m \in M} d_g^2(g(m),h(m))d\mu(m) = \int_{m \in M} \|\operatorname{exp}_{g(m)}^{-1}(h(m))\|^2d\mu(m). $$

Fix some non-constant $f \in L^2(\mu;M,m_0)$.

Let $c\in M$ and let $C\subseteq L^2(\mu;M,m_0)$ be proper and non-empty. Under what joint conditions on $C$ and $c$ can we guarantee that $$ \operatorname{argmin}_{g \in C} D(f,g) = \operatorname{argmin}_{g \in C} \int_{m \in M} \|\exp_{c}^{-1}(f(m))-\operatorname{exp}_{c}^{-1}(g(m))\|^2d\mu(m) . $$

Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of all equivalence classes of measurable functions $g:X\rightarrow M$ with finite the distance $D(m_0,g)$ from the constant function $m_0$ where $D$ is the metric: $$ D(g,h):=\int_{m \in M} d_g^2(g(m),h(m))d\mu(m) = \int_{m \in M} \|\operatorname{exp}_{g(m)}^{-1}(h(m))\|^2d\mu(m). $$

Fix some non-constant $f \in L^2(\mu;M,m_0)$.

Let $c\in X$ and let $C\subseteq L^2(\mu;M,m_0)$ be proper and non-empty. Under what joint conditions on $C$ and $c$ can we guarantee that $$ \operatorname{argmin}_{g \in C} D(f,g) = \operatorname{argmin}_{g \in C} \int_{m \in M} \|\exp_{c}^{-1}(f(m))-\operatorname{exp}_{c}^{-1}(h(m))\|^2d\mu(m) . $$

Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of all equivalence classes of measurable functions $g:X\rightarrow M$ with finite the distance $D(m_0,g)$ from the constant function $m_0$ where $D$ is the metric: $$ D(g,h):=\int_{m \in M} d_g^2(g(m),h(m))d\mu(m) = \int_{m \in M} \|\operatorname{exp}_{g(m)}^{-1}(h(m))\|^2d\mu(m). $$

Fix some non-constant $f \in L^2(\mu;M,m_0)$.

Let $c\in X$ and let $C\subseteq L^2(\mu;M,m_0)$ be proper and non-empty. Under what joint conditions on $C$ and $c$ can we guarantee that $$ \operatorname{argmin}_{g \in C} D(f,g) = \operatorname{argmin}_{g \in C} \int_{m \in M} \|\exp_{c}^{-1}(f(m))-\operatorname{exp}_{c}^{-1}(g(m))\|^2d\mu(m) . $$