4 Point 4 re-framed

I come from a background of having done undergraduate and graduate courses in General Relativity and elementary course in riemannian geometry.

Jurgen Jost's book does give somewhat of an argument for the the statements below but I would like to know if there is a reference where the following two things are proven explicitly,

1. That the sectional curvature of a 2-dimensional subspace of a tangent space at a point on the Riemannian manifold is independent of the choice of basis. That is the definition of the sectional curvature depends only on the choice of the 2-dim subspace.

2. That the sectional curvature determines the Riemannian curvature fully.

Secondly can one give me a reference where I can see how in practice is sectional curvature computed. To a first timer to this subject it is not obvious how one does a calculation on "all" 2-dimensional subspaces of a high-dimensional space. Especially when people talk of manifolds with "constant sectional curvature". How are they realized?

I would like to see some explicit examples to understand this point.

Further some studies about homogeneous spaces (needed to understand some issues in Quantum Field Theory) got me to the following 4 very non-trivial ideas in Riemannian manifolds which I am stating in my own way here ,

1. That the isometry group of a Riemannian manifold is always a lie group.

2. The isotropy subgroup of any point on a Riemannian manifold under the smooth transitive action of its own isometry group on itself is a compact subgroup. (The context being what is called a "Riemannian Homogeneous Space")

3. {This point was earlier framed in a way which made the bi-implication false as pointed out by some people} The formulation should be as follows.

A Riemannian Homogeneous Space is a riemannian manifold on which the isometry group acts transitively. Now the theorem is that such a space is compact IFF its isometry group is compact.

Thats the statement whose intuition I am looking for.

Apologies for the confusion caused.

4. { This question too was not framed properly. Basically I could not figure out how to write the nabla for the connection! It should be as Jose has pointed out.}

A Riemannian riemannian manifold is locally symmetric IFF the connection acting on if and only if the scalar Riemann curvature gives 0tensor is parallel with respect to the Levi-Civita connection.

Can one give me the intuition behind these or give me specific references where these are proven in explicit details?

3 Point 3 has been corrected

I come from a background of having done undergraduate and graduate courses in General Relativity and elementary course in riemannian geometry.

Jurgen Jost's book does give somewhat of an argument for the the statements below but I would like to know if there is a reference where the following two things are proven explicitly,

1. That the sectional curvature of a 2-dimensional subspace of a tangent space at a point on the Riemannian manifold is independent of the choice of basis. That is the definition of the sectional curvature depends only on the choice of the 2-dim subspace.

2. That the sectional curvature determines the Riemannian curvature fully.

Secondly can one give me a reference where I can see how in practice is sectional curvature computed. To a first timer to this subject it is not obvious how one does a calculation on "all" 2-dimensional subspaces of a high-dimensional space. Especially when people talk of manifolds with "constant sectional curvature". How are they realized?

I would like to see some explicit examples to understand this point.

Further some studies about homogeneous spaces (needed to understand some issues in Quantum Field Theory) got me to the following 4 very non-trivial ideas in Riemannian manifolds which I am stating in my own way here ,

1. That the isometry group of a Riemannian manifold is always a lie group.

2. The isotropy subgroup of any point on a Riemannian manifold under the smooth transitive action of its own isometry group on itself is a compact subgroup. (The context being what is called a "Riemannian Homogeneous Space")

3. That isometry group of

4. {This point was earlier framed in a way which made the bi-implication false as pointed out by some people} The formulation should be as follows.

A Riemannian Homogeneous Space is a riemannian manifold on which the isometry group acts transitively. Now the theorem is that such a space is compact IFF the Riemannian manifold its isometry group is compact.

Thats the statement whose intuition I am looking for.

Apologies for the confusion caused.

5. A Riemannian manifold is locally symmetric IFF the connection acting on the scalar curvature gives 0.

Can one give me the intuition behind these or give me specific references where these are proven in explicit details?