It is difficult to reconcile your first two statements, because they are actually wrong as written!

A riemannian manifold is locally symmetric if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection. This condition was studied by Élie Cartan, who classified them using his classification of real semisimple Lie algebras. My favourite reference for this is Besse's Einstein manifolds, Chapter 7F, from where I quote the following theorem:

10.72 Theorem (Elie Cartan). For a Riemannian manifold (not necessarily complete) the following conditions are equivalent:

 

(i) $DR = 0$;

 

(ii) the geodesic symmetry $s_p$ around any point $p$ (which is defined only locally) is an isometry.

 

For a complete Riemannian manifold $(M,g)$ the two following conditions are equivalent:

 

(iii) for every point $p$ in $M$ the geodesic symmetry around $p$ is well defined and is an isometry;

 

(iv) the manifold $M$ is a homogeneous space $G/H$ where $G$ is a connected Lie group, $H$ a compact subgroup of $G$, and where there exists an involutive automorphism $\sigma$ of the group $G$ for which, if $S$ denotes the fixed point set of $\sigma$ and $S_e$ its connected component of the identity one has $S_e \subset H \subset S$. Moreover the Riemannian metric under consideration on $G/H$ is invariant under $G$.

 

Furthermore: if $(M, g)$ satisfies (iii) or (iv), then it satisfies (i) and (ii). If $(M, g)$ satisfies (i) or (ii) and if it is simply connected and complete then it satisfies (iii) and (iv).

 

10.76 Definition. A Riemannian manifold is said to be locally symmetric if it satisfies (i) or (ii) above; it is said to be symmetric if it satisfies (iii) or (iv).

I suspect the source of the confusion might be with the meaning of "constant curvature", which usually means constant sectional curvature (or perhaps parallel Riemann curvature) and not constant scalar curvature, which is a much weaker condition.

Concerning your final parenthetic question, there is a theorem of Ambrose and Singer, reformulated by Kostant, which says that a riemannian manifold $(M,g)$ is locally homogeneous if and only if it admits a metric connection with parallel torsion and parallel curvature. It is locally symmetric if (and only if) the connection is torsion-free.

It is difficult to reconcile your first two statements, because they are actually wrong as written!

A riemannian manifold is locally symmetric if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection. This condition was studied by Élie Cartan, who classified them using his classification of real semisimple Lie algebras. My favourite reference for this is Besse's Einstein manifolds, Chapter 7F, from where I quote the following theorem:

10.72 Theorem (Elie Cartan). For a Riemannian manifold (not necessarily complete) the following conditions are equivalent:

(i) $DR = 0$;

(ii) the geodesic symmetry $s_p$ around any point $p$ (which is defined only locally) is an isometry.

For a complete Riemannian manifold $(M,g)$ the two following conditions are equivalent:

(iii) for every point $p$ in $M$ the geodesic symmetry around $p$ is well defined and is an isometry;

(iv) the manifold $M$ is a homogeneous space $G/H$ where $G$ is a connected Lie group, $H$ a compact subgroup of $G$, and where there exists an involutive automorphism $\sigma$ of the group $G$ for which, if $S$ denotes the fixed point set of $\sigma$ and $S_e$ its connected component of the identity one has $S_e \subset H \subset S$. Moreover the Riemannian metric under consideration on $G/H$ is invariant under $G$.

Furthermore: if $(M, g)$ satisfies (iii) or (iv), then it satisfies (i) and (ii). If $(M, g)$ satisfies (i) or (ii) and if it is simply connected and complete then it satisfies (iii) and (iv).

10.76 Definition. A Riemannian manifold is said to be locally symmetric if it satisfies (i) or (ii) above; it is said to be symmetric if it satisfies (iii) or (iv).

I suspect the source of the confusion might be with the meaning of "constant curvature", which usually means constant sectional curvature (or perhaps parallel Riemann curvature) and not constant scalar curvature, which is a much weaker condition.

Concerning your final parenthetic question, there is a theorem of Ambrose and Singer, reformulated by Kostant, which says that a riemannian manifold $(M,g)$ is locally homogeneous if and only if it admits a metric connection with parallel torsion and parallel curvature. It is locally symmetric if (and only if) the connection is torsion-free.

It is difficult to reconcile your first two statements, because they are actually wrong as written!

A riemannian manifold is locally symmetric if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection. This condition was studied by Élie Cartan, who classified them using his classification of real semisimple Lie algebras. My favourite reference for this is Besse's Einstein manifolds, Chapter 7F.

I suspect the source of the confusion might be with the meaning of "constant curvature", which usually means constant sectional curvature (or perhaps parallel Riemann curvature) and not constant scalar curvature, which is a much weaker condition.

Concerning your final parenthetic question, there is a theorem of Ambrose and Singer, reformulated by Kostant, which says that a riemannian manifold $(M,g)$ is locally homogeneous if and only if it admits a metric connection with parallel torsion and parallel curvature. It is locally symmetric if (and only if) the connection is torsion-free.

It is difficult to reconcile your first two statements, because they are actually wrong as written!

A riemannian manifold is locally symmetric if and only if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection. This condition was studied by Élie Cartan, who classified them using his classification of real semisimple Lie algebras. My favourite reference for this is Besse's Einstein manifolds, Chapter 7F, from where I quote the following theorem:

10.72 Theorem (Elie Cartan). For a Riemannian manifold (not necessarily complete) the following conditions are equivalent:

(i) $DR = 0$;

(ii) the geodesic symmetry $s_p$ around any point $p$ (which is defined only locally) is an isometry.

For a complete Riemannian manifold $(M,g)$ the two following conditions are equivalent:

(iii) for every point $p$ in $M$ the geodesic symmetry around $p$ is well defined and is an isometry;

(iv) the manifold $M$ is a homogeneous space $G/H$ where $G$ is a connected Lie group, $H$ a compact subgroup of $G$, and where there exists an involutive automorphism $\sigma$ of the group $G$ for which, if $S$ denotes the fixed point set of $\sigma$ and $S_e$ its connected component of the identity one has $S_e \subset H \subset S$. Moreover the Riemannian metric under consideration on $G/H$ is invariant under $G$.

Furthermore: if $(M, g)$ satisfies (iii) or (iv), then it satisfies (i) and (ii). If $(M, g)$ satisfies (i) or (ii) and if it is simply connected and complete then it satisfies (iii) and (iv).

10.76 Definition. A Riemannian manifold is said to be locally symmetric if it satisfies (i) or (ii) above; it is said to be symmetric if it satisfies (iii) or (iv).

I suspect the source of the confusion might be with the meaning of "constant curvature", which usually means constant sectional curvature (or perhaps parallel Riemann curvature) and not constant scalar curvature, which is a much weaker condition.

Concerning your final parenthetic question, there is a theorem of Ambrose and Singer, reformulated by Kostant, which says that a riemannian manifold $(M,g)$ is locally homogeneous if and only if it admits a metric connection with parallel torsion and parallel curvature. It is locally symmetric if (and only if) the connection is torsion-free.