Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$.

Consider the first nonzero eigenvalue equation on the traceless-transverse symmetric $2$-form $h$ on $S^3/\Gamma$, that is $$ \Delta h+\lambda_1 h=0, $$ where $\text{div}_gh=\text{Tr}_g h=0$. It is well-known that if $\Gamma$ is trivial, then $\lambda_1=6$.

Conversely, if one already knows $\lambda_1=6$, can we have a classification of $\Gamma$? I am particularly interested in the case of whether $\Gamma$ is conjugate to a subgroup of $U(2)$.

Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$.

Consider the first nonzero eigenvalue equation on the traceless-transverse symmetric $2$-form $h$ on $S^3/\Gamma$, that is $$ \Delta h+\lambda_1 h=0, $$ where $\Delta$ denotes the rough Laplacian and $\text{div}_gh=\text{Tr}_g h=0$. It is well-known that if $\Gamma$ is trivial, then $\lambda_1=6$.

Conversely, if one already knows $\lambda_1=6$, can we have a classification of $\Gamma$? I am particularly interested in the case of whether $\Gamma$ is conjugate to a subgroup of $U(2)$.

Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$.

Consider the first nonzero eigenvalue equation on the traceless-transverse symmetric $2$-form $h$ on $S^3/\Gamma$, that is $$ \Delta h+\lambda_1 h=0, $$ where $\text{div}_gh=\text{Tr}_g h=0$. It is well-known that if $\Gamma$ is trivial, then $\lambda_1=6$.

Conversely, if one already knows $\lambda_1=6$, can we have a classification of $\Gamma$?

Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$.

Consider the first nonzero eigenvalue equation on the traceless-transverse symmetric $2$-form $h$ on $S^3/\Gamma$, that is $$ \Delta h+\lambda_1 h=0, $$ where $\text{div}_gh=\text{Tr}_g h=0$. It is well-known that if $\Gamma$ is trivial, then $\lambda_1=6$.

Conversely, if one already knows $\lambda_1=6$, can we have a classification of $\Gamma$? I am particularly interested in the case of whether $\Gamma$ is conjugate to a subgroup of $U(2)$.